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*x *E *y for all x, y E V. 2 In this case we say that *v is an extension of v via * ; {2) a standard core embedding, or an €-isomorphism of v onto the standard core of *v, if in addition *v = ( *V ; *E, *st) is a st-e-structure and the standard core s < •v) = {z E *V : "'s t z} coincides with {*x : X E V}. In this case we say that *v is a standard core extension of v via * ; 1 By the reasons related to Theorem 1.5.11 there will be no need for invariant structures in this Chapter. 2 Notation: ·x is typically used instead of *(X) .
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{3) an elementary €-embedding if we have cpv
(*tpfv whenever cp is a closed E-formula with sets in V as parameters and V; is obtained by the substitution of *x for any x E V occurring in cp as a parameter. In this 0 case we say that *v is an elementary extension of v via * .
Exercise 4.1.2. Let *v = ( "'V ; "'€, "'st) be a standard core extension of v = ( V ; E). Prove that if cp and *tp are as in 4.1.1{3) then cp is true in v iff ( V;) st 0 is true in *v. 4.1b Nonstandard extensions of theories By a standard set theory we understand a theory in a language which includes the E-language but does not contain st . A nonstandard set theory will be a theory in a language which includes the st-e-language. Thus ZFC is a standard theory while HST, BST, 1ST are nonstandard theories. Any nonstandard theory � distinguishes the standard core S = { x : st x } of the set universe, and hence for any e-formula cp, cp8 t is a formula of the language of �- The next definition introduces several important notions which characterize the relationships between a nonstandard theory � and a standard theory 11, in terms of the standard core of the universe of �. Definition 4.1.3. (1) � is a standard core extension of a theory 11 in the e-language if for any axiom � of 11, �s t is a theorem of �- 3 (2) � is a conservative standard core extension of 11 if for any e-formula �, 11 proves � if and only if � proves �st . 3 (3) � is a reducible nonstandard theory if for any sentence � of the language of � there is an e-sentence 1/J such that � proves � 1/Js t . {4) � is standard core interpretable in 11 if there exist: 1) an interpretation *v = ( "'V ; *E , "'s t, .. . ) of � in 11, where "'st interprets the atomic predicate st while ... denotes classes which interpret other possible atomic sy�bols of the language of �, and 2) a standard core embedding * : V � � of the 11-universe v = ( V ; e) of all sets into *v. 4 Such an interpretation is called: a standard core interpretation. 0 Remark 4.1.4. The notion of conservativity in 4.1.3{2) is different from Definition 1.5.16. Yet both notions obviously coincide for nonstandard theo 0 ries containing I n ner Transfer like BST or 1ST . 3 In (1), (2) 11 is supposed to be a standard theory in the E-language like ZFC. If 11 is a theory in a language properly extending the E-language, like ZFC19 of § 1 .5f or theories with global choice like ZFGT below, then the definitions become more complicated, see Remark 4.6.13 on a suitable example. 4 As usual, both the interpretation and the embedding must be defined by formulas of the language of 11, and their indicated properties provable in 11 .
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Remark 4.1.5. Nonstandard theories with a meaningful class WIF of well founded sets, like HST, admit modifications of these definitions oriented towards WIF rather than S. A nonstandard theory � is a conservative wf core extension of 11 if, for any e-sentence �, 11 proves � iff � proves �vf . The notion of a wf-core intef,retable theory, along with an associated notion of the well-founded core WIF V} of a st-E-structure *V is defined similarly. For 'I = HST, these concepts are equivalent to the standard core notions 0 since the classes S and WIF are €-isomorphic in HST . Standard theories may transcend ZFC by both new axioms (like the continuum-hypothesis) and new elements of the language (so does ZFC19 of § 1.5f or theories with global choice in Definition 4.3.4), in both cases the set universe satisfies ZFC. The language of a standard core extension of ZFC is at least the st-E-language, hence, a standard universe (core) S = {x : st x} is defined, and the requirement 4.1.3{1) means that ( S ; E) is postulated, by �, to interpret 11. Standard core interpretability of a nonstandard theory � in a standard theory 11 means that 11 is strong enough to define a structure that interprets � in 11 (§ 1.5d), along with an isomorphism of the universe of all sets onto the standard core of the structure - and thus the set universes of both theories must be connected in a certain way, in addition to the general requirements contained in the definition of interpretation in § 1.5d. Many examples of inter pretation of nonstandard theories in ZFC (and some other standard theories) will be given below. The properties of conservativity, reducibility, interpretability, together with equiconsistency with ZFC, will be the main issues of the metamath ematical study of nonstandard theories below. The following definition in troduces a property of structures rather than theories, but still it contains a certain indirect characterization of nonstandard set theories.
Definition 4. 1.6 (in ZFC ). Let 'I be a theory in the st-E-language. A set M is �-extendible if ( M ; E f M) admits a standard core embedding into a st-E-structure which models �0 Exercise 4.1.7. {1) Prove that every conservative standard core extension of ZFC is equiconsistent with ZFC. (Reducibility does not imply equicon sistency, moreover, every inconsistent extension of ZFC is reducible.) (2) Prove that if 'I is a standard core extension of ZFC then any �extendible set is a model of ZFC. 0 4. 1c Comments Why are these properties important and deserve attention besides just an interest related to a purely foundational study ? Suppose that one is going to "work" in a nonstandard set theory �, that is, to prove theorems in � and
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interpret the results as mathematically true. Naturally, � is a theory whose language contains both E and st The standard core S = { x : st x } of the set universe of � can be identified with the convenional mathematical set universe 5 • Since the legitimate kit of mathematical tools is almost universally identified with ZFC, it is reasonable to require that the theory � proves those and only those €-statements about standard sets which ZFC proves about all sets, which is exactly the standard core conservativity requirement. Objects outside of S can be viewed in two ways. We can see them as auxi liary objects which do not possess the same mathematical reality as those in S, or, saying it differently, as objects invented by �, which appear in the beginning of a proof and die with QED, when we forget about them. However, if mathematics is not merely a formal game for us we should consider it as a principle that the "nonstandard" objects ought to have some kind of reality too, perhaps a "relativized" reality with respect to S which is taken as "real" . Then � would only provide a kind of �-envelope of S. In this situation we may want the "envelope" to fit tightly to S such that all st-E-properties sets in S do have in the envelope are traceable down to S. This is where the property of reducibility appears. But at the end of the day a direct definition of the "envelope" within S is the best thing ! (Compare with the definition of complex numbers as pairs of reals.) Here we face an obstacle: it is literally impossible to extend the universe of all sets since everything is already here. This is where the notion of interpretable extension appears: standard core interpretability of � in 11 means, informally, that a theory 11 is strong enough to extend the universe V of all sets to a structure satisfying � where V becomes the class of all standard sets. The distinguished role of ZFC in the foundations of "standard" mathe matics leads us to the following definition: .
Definition 4.1.8. A nonstandard set theory is "realistic" 6 iff it admits a standard core interpretation in ZFC. 0 We consider the property of being "realistic" as a principal property which separates nonstandard theories that reflect mathematical reality (as long as the latter is based on the Zermelo - Fraenkel system ZFC ) from schemes of a purely syntactical nature. It will be our goal to prove that amongst the nonstandard theories considered in this book, BST and HST are "realistic" while 1ST and some theories considered in Chapter 8 are not.
Proposition 4.1.9. Any "realistic" nonstandard theory � is a conservative (hence, equiconsistent) standard core extension of ZFC . 5 A special feature of HST is that it allows to consider the class WIF of well-found
ed sets, an E-isomorphic, and transitive copy of S as a more convenient domain of objects of "standard" mathematics than S. Note that Nelson considers things differently, see Footnote 2 on page 13. 6 The meaning of this word here is not the same as in Hrbacek [Hr 01 ) .
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Proof. That conservativity implies equiconsistency is easy: if � proves 0 = 1 then {0 = 1) s t is also provable, thus ZFC proves 0 = 1. To establish conservativity, suppose that � proves �s t , where � is an €-sentence. We have to prove � in ZFC. Consider a standard core interpretation q of � in ZFC. Then �st is true in q, and hence � is true in the standard core of q . Therefore, as the latter is €-isomorphic to the ZFC universe V , � i s true in V as well, which is a proof of � in ZFC . 0 4.1d Metamathematics of internal theories: the main results Claim {iv) of the next theorem involves ZFGT, a theory {defined in § 4.6a) in the language Ce,G,T with symbols G for a global choice function and T for the truth predicate for formulas of Ce,G · Note that ZFGT contains Separation in Ce,G,T, but the schemata of Replacement and Collection are included in the €-language only. It will be shown {Theorem 4.6.3) that ZFGT is a conservative {in the sense of Definition 1.5.16) standard extension of ZFC .
Theorem 4.1.10. {i) BST is a "realistic" theory - hence, it is an equiconsistent and conservative standard core extension of ZFC . (ii) BST is a reducible theory - this follows from Theorem 3.2.3{ii). (iii) 1ST is an equiconsistent and conservative standard core extension of ZFC. However 1ST is not a reducible theory and 1ST is not standard core interpretable in ZFC - hence it is not "realistic". (iv) On the other hand 1ST is standard core interpretable in ZFGT . Claim (i) of Theorem 4.1.10 will be established in § 4.3c, claims (iii), {iv) related to 1ST - in Sections 4.4, 4.5, 4.6. Note that the conservativity and equiconsistency of BST in {i) easily follow from these properties of 1ST via the inner model of bounded sets, but the standard core interpretability of BST does not seem to follow from any property of 1ST whatsoever. Theorem 4.1.10, in its BST part, will be an essential precondition in our study of metamathematical properties of HST in Chapter 5. Applying the conservativity in Theorem 4.1.10 and Inner Transfer, we have Corollary 4.1.11. Any of the four following conditions is necessary and sufficient for an €-sentence � to be a theorem of ZFC : (1) � is a theorem of BST ; {3) �st is a theorem of BST ; 0 {2) � is a theorem of 1ST ; (4) �st is a theorem of 1ST . Let us draw several further consequences. Claims {ii) and (iii) in the next corollary is our backlog from § 3.4a: {iii) and {iv) of Theorem 3.4.5. They show that BST is the theory of the class 18 of all bounded sets in 1ST, while BST 7 is the theory of classes D" in 1ST. The claims can also be viewed as a sort of conservativity of 1ST over BST and of BST over BSTK. K
7 See § 3.3a on partially saturated theories BST� and BST��: .
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Corollary 4.1.12. (i) BSTK is a "realistic" theory - and hence it is an equiconsistent and conservative standard core extension of ZFC . 8 (ii) H � is a st-E-sentence then BST proves � iff 1ST proves �bd . (iii) If � is a st-E-sentence then BSTK proves � iff BST proves that �s iC {the relativization to n� ) holds for any standard infinite cardinal "' ·
Proof. {i) Recall that by Theorem 3.4.5(i) any structure of the form 0� = ( 0� ; E , st, K-) , where "' is an infinite standard cardinal, is an interpretation of BSTK in BST. (Note: the cardinal "' is an interpretation of the constant "' of the language of BSTK .) Obviously 0� contains all standard sets. Taking, for instance, "' = No {or N 1 , etc.), we find the following: any standard core inter pretation of BST in ZFC can be reduced to a standard core interpretation of BSTK in ZFC. 9 It remains to apply Theorem 4.1.10{i). {ii) The direction "only if" follows from (i) of Theorem 3.4.5: indeed if � is a theorem of BST then it must be true in 18 because this class interprets BST. To establish the claim "if" , suppose that 1ST proves �bd where, we recall, bd indicates relativization to the class 18 = { x : 3st y ( x E y)} of all bounded sets in 1ST. By Theorem 3.2.3{ii) {Reduction to E2t ) there is an €-sentence cp such that BST proves cp �. As 18 is an interpretation of BST in 1ST by Theorem 3.4.5{i), and 1ST proves �bd , 1ST also proves cpbd , and hence proves cp8 t by Inner Transfer of BST. Then ZFC proves cp by Corollary 4.1.11, thus BST proves cp8t and cp itself by ZFCst and Inner Transfer of BST. It follows that BST proves � by the choice of cp . (iii) B y Theorem 3.4.5{ii), we can concentrate on the claim "if" . We shall assume that Card indicates only infinite cardinals in the course of the proof. Suppose that BST proves ys t "' E Card �0 -c . It follows from Theorem 3.3.5 (Reduction to E2t in BSTK ) that there is an E-formula cp(K-) containing the constant "' such that BSTK proves � cp{K-). Then, as 0� is an interpretation of BSTK in BST by Theorem 3.4.5{ii), BST also proves V "' E Card cp{K-) 0 -c and hence proves V "' E Card cp(K-) by I nner Transfer of BST and of BSTK. But then ZFC proves V "' E Card cp(K-) by the conservativity of BST and BSTK proves V "' E Card cp(K-) by the conservativity of BSTK. (I nner Transfer also works in this argument.) However BSTK postulates "' to be an infinite cardinal. It follows that BSTK proves cp("'), and hence proves � by the choice of cp . 0 Exercise 4.1.13. Study the reducibility of BSTK. In this case ZFC cannot serve as a ground standard theory because Reduction to E2t in Theorem 3.3.5 leads to E-formulas containing "' However we can enrich ZFC by "' as a constant, with an associated axiom saying that "' is an infinite cardinal. 0 ·
8 Proposition 4.3.2 below proves that BST�, the other partially saturated theory, 9
is also "realistic" , in a somewhat modified sense. Yet no result like 4 . 1.12(iii) is known for BST�. See also § 6.2b. A more direct construction of a standard core interpretation of BSTK in ZFC is outlined in Exercise 4.3. 15.
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4.2 Ultrapowers and saturated extensions
Technically, the proof of Theorem 4.1.10 will consist of a series of nonstandard extensions of different standard structures. In this section we review the basic tools involved. The focal point will be the saturation properties of extensions. We begin in § 4.2a with a brief introduction into saturated extensions and enlargements as nonstandard structures; Theorem 4.2.4 will show which parts of nonstandard theories are satisfied in these nonstandard structures. All nonstandard extensions used below belong to a certain general cate gory called quotient powers in § 4.2b. This class includes ordinary ultrapow ers, ultralimits, iterated and "definable" ultrapowers in a uniform and natural fashion. Then we consider two particular classes of ultrafilters in § 4.2c, ade quate and good ultrafilters: they naturally lead to saturated quotient powers. Limits of transfinite elementary chains of extensions are considered in § 4.2d. 0 Blanket agreement 4.2. 1 . We argue in ZFC in this section.
4.2a Saturated structures and nonstandard set theories The property of saturation is considered here in less generality than in model theory but more in line with its applications in this book. See 4.2.5 for a more general concept. Definition 4.2.2. Let "' be an infinite cardinal. An €-structure ( *V ; *E) is K--saturated iff any family !C � *V with card !C < "' and satisfying *£-f. i. p. (the finite intersection property w. r. t. *E, meaning that any finite subfamily !C' � !C has a common *£-element) in ( *V ; *E) has an *£-element in *V common for the whole family !C . Suppose that (V ; E) is another €-structure and * : V -+ *V is an € embedding. Then ( *V ; *E) is a K--enlargement of (V ; E) via * iff any family !C � {*X : X E V} with card !C < K-, *£-f. i. p. in ( *V ; *E) , has an *£ element in *V common for the whole family !C . 0 Exercise 4.2.3. Suppose that * : V -+ *V is an €-embedding of an € structure v = (V ; E) in *v = ( *V ; *e) . Prove the following: (1) if *v is K--saturated then it is a K--enlargement of v via * ; (2) for *v to be a K--enlargement of v via * the following is necessary and, in the case when * is an elementary embedding, also sufficient: for any family !C � V with card !C < K-, satisfying e- f. i. p. in v, the family 0 *!C = {*X : X E !C} has a common *£-element in *V . Thus the property of K--enlargement essentially requires that any f. i. p. family of size < "' in the original structure gains an element in the extension. The next theorem contains sufficient conditions for a standard core ex tension of a standard structure to satisfy certain axioms. Recall that K--size Bl is Basic Idealization of § 3.1b in the case card A0 � "" (see § 3.3a). Similarly let K--size BE be Basic Enlargement of § 3.1b in the case card A0 � "" ·
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Theorem 4.2.4. Suppose that V is either a set of the form ViJ ,
{) be ing a limit ordinal, or else the universe V of all sets, and * is an ele mentary standard core embedding of v = ( V ; E t V) in a st-E-structure *v = ( *V ; *E , *st) . Let finally "' E V be a cardinal in V. 10 Then :
(i) *v satisfies 1 1 Inner Transfer, Inner Standardization, ze, and ze st ; {ii) if v satisfies ZFe then *v satisfies ZFe st ; {iii) if for any x E *V there is an element a E V such that x *e *a then *v satisfies Inner Boundedness ;
(iv) if there exists � E *V such that � � *"- and for any x E *V there is a function f E V defined on "' such that x = 1(€) then *v satisfies Inner Strong K--Boundedness (see § 3.3a) ; (v) if *v is "'+-saturated then it satisfies K--size 81 ; (vi) if *v is a "'+ -enlargement of v then it satisfies K--size BE .
Proof. (i) and (ii) . Inner Transfer follows from the result of Exercise 4.1.2 and the elementarity of the embedding * · ze st holds by the same reasons, and also because v itself obviously satisfies ze {any set v{) , {) limit, does). To check Inner Standardization, let rp(x) be an E-formula with sets in *V as parameters. Suppose that X E V; then *X is a standard set in *v while Y = {x E X : cjv(*x) } is a set in V by the choice of V. Thus *Y E *V is a standard set in *v. We claim that yst x E *X (x E *Y - rp(x)) holds in *v. Since standard sets in *v are those of the form *y, it suffices to show *x *E *Y - rp•v { *x ) for every x E X. Yet either side is equivalent to x E Y. (iii) is obvious. {iv) follows from Lemma 3.3.2{ii) (v) The notion of finiteness is obviously absolute for V. Thus any set B E *V such that "B is standard and finite" holds in *v has the form B = i1. for a unique finite A E V. This observation reduces K--size 81 in *v to yfin A � Ao 3 x *e *X V a E A (x *e Xa)
3 x *e *X V a E Ao (x *e Xa) {t)
where A and Xo are sets in V, card Ao � K-, and Xa E *V for any a E A. {The sets Xa arise as follows. Take, as in Basic Idealization, 1/J E *V such that "1/J is a map i1.0 -+ &(*X) " holds in *v. Let Xa be the unique element of *V such that 1/J(*a) = Xa holds in *v. The argument is validated by the fact that *v satisfies ze by (i) .)
10 11
That is, "' E Ord and V does not contain a bijection from K onto any e < K . The word "satisfies" in the theorem means either models, in the form v f=
Y E U whenever X � Y � I) and not containing 0. An ultrafilter is a filter U containing exactly one element of any pair of complementary sets {classes) X, I ' X in &(I). In those cases below when I is a proper class, accordingly, U is a collection of proper classes, suitable provisions will be taken to fix a parametrization by sets of the classes involved, to keep the arguments within legitimate frameworks. 12 Classes in ZFC are collections defined by formulas, like e. g. Ord. We'll have V = V, the ZFC universe of all sets, in the most important applications.
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The following requirements 3°, 4° will be instrumental in the proof of the Los theorem below. Note that both requirements are satisfied for obvious reasons provided U is an ultrafilter and § = V1 . 3°. If /1 , ... , /n E § and rp(x1 , ... , xn ) is an E-formula then the set X = {i E I : rpv (/1 (i) , ... fn(i} } is U-measurable, i. e. it belongs to U or to the complementary ideal U = {I ' X : X E U} . 4° . If /1 , ... , /n E §, rp{x, x1 , ... , xn) is an E-formula, and 1/J(x1 , ... , xn) is 3 x rp(x, x1 , ... , Xn) then Vi E I 1/Jv(/1 {i} , ... , /n(i}} => 3 f E § Vi E I rpv {/(i}, /1 {i} , ... , /n(i)} .
J
Recall that rpv means that the E-formula rp is relativized to v, that is all quantifiers 3 z, V z are replaced by 3 z E V, V z E V and E changed to E . In different words rpv means that rp is true in v = ( V ; E) , see § 4.la. If V is a set rather than a proper class then rpv can be replaced by v t= rp .
Definition 4.2.6. If U, I are as indicated then Ui 4>(i), Ui E I 4J(i} mean that the set {i E I : 4>{i} } belongs to U. {The quantifier: "U-many" . ) 0 Under the assumptions 1 ° - 4° put for all /, g E §
f *= g iff Ui {/(i} = g(i}} ; f *e g iff Ui {/(i) E g(i)} .
*st f iff f *= f:z: for some x E V ;
Exercise 4.2. 7. Prove that *= is an equivalence relation on § and the 0 relations \: and *st are *=-invariant. Thus ( § ; *E, *st ; *= ) is an invariant structure in the sense of § 1.5c. Yet we are more interested in the associated quotient structure (§/ *= ; *E, *st) .
Definition 4.2.8. Put [/] = {g E § : f *= g} for any f E §, and further
[/] \: [g] iff f *e g , *st [/] iff *st f ,
*x = [f:z:]
for each x E V,
*V = §I ( *= ) = {[/] : f E §} .
If V is a proper class - a rather typical case below - then the definition of [/] is to be amended so that the classes [/] become sets. We define [/] = {g E § n Vo:(J) : f *= g} for any f E $, where o:(f) is the least ordinal o: such that V0 , the von Neumann set, contains some g E § with f *= g. The structure *v = ( § /*= ; \:, *st) = ( *V ; \:, *st), also denoted by § / U and often truncated to ( *V ; \:) , is the U, §-quotient power of v = ( V ; E) . The map x � *x : V -+ *V is the natural embedding. If I is a set then the quotient power is called set-indexed. 0
Exercise 4.2.9. Prove that {*x : x E V} is the standard core of *v in the sense of § 4. 1a, that is the collection of all *st-standard elements of * V. o
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Clearly ( *V ; *e) is a usual ultrapower of ( V ; e) provided U is an ultra filter and § = V1 . Conditions 3° and 4° are obvious in this case. We need another definition to formulate the Los theorem. If � is any € formula � with parameters in § then for any i E I we let �[i] indicate the result of the substitution of f (i) for every f E § in �, and let [�] indicate the result of the substitution of [/] for every f E § in �- Thus �[i] and [�] are E-formulas having sets in resp. V and *V as parameters.
Lemma 4.2.10 (Los Theorem) . Under the assumptions 1° - 4°, suppose that � is a closed €-formula with parameters in §. Then [�fv Ui {�v [i]). Proof. We argue by induction on the length of �- The case of elementary formulas f = g, f E g easily follows from the definition. It suffices to con sider only the induction steps for A, ..., , 3 . The step for A is trivial. The step for ..., follows from the equivalence Ui ..., �[i] ..., Ui �[i] by standard ar guments. The equivalence itself is a consequence of 3° . The step for 3 . Prove the lemma for a formula "CJ! := 3 x �(x) assuming that the result holds for �(/) whenever f E §. The direction => is trivial: ["CJ!fv implies [�(/)fv for some f E §, hence Ui �v(/)[i] by the induction hypothesis. This obviously implies Ui "CJ!v[i]. The direction . ) is the direct limit of ( � ; Ed , � < A, in the sense that V>. = {e� >. (x) : � < A A x E � } . 0 Note that any initial segment {} � Ord+ is either an ordinal or the class Ord of all ordinals or Ord+ itself.
Proposition 4.2.19. Suppose that ( � ; Ed and e71� : V71 -+ � (TJ � � � 'Y) is an elementary continuous chain of €-structures, 'Y being either a limit ordinal or oo. If "' is a cardinal, cof 'Y > K-, and any ( � + 1 ; E� +1 ) is a K--enlargement of ( � ; E�) then ( V-y ; E-y) is K--saturated. Proof. Consider any E-y-f. i. p. set !r � V-y of cardinality < "-· (Recall that E-y-f. i. p. indicates the finite intersection property w. r. t. E� as the member ship, see Definition 4.2.2.) Since cof 'Y > K-, there is an ordinal � < "' such that every X E !r has the form X = e11-y (Y) for some TJ < � and Y E V71• In this case the set Z = e71� (Y) belongs to � and still X = e�-y(Z) . It follows that there is a set !Z � � still with card !Z = card !r < "' such that !r = { e�-y{Z) : z E !Z}. As e�"Y is an elementary embedding, :& is E� -f. i. p . . However ( �+ 1 ; E� + 1 ) is a K--enlargement of ( � ; E� ) , and hence there is an element z E �+ 1 with z E� + 1 e �,� + 1 (Z) for any Z E !&. Then x = e� + 1 ,-y{z) satisfies x E e�-y(Z) for any Z E !Z simply because e�-y(Z) = e� + 1 ,-y{e� ,� + 1 (Z)) . 0 There are different methods to maintain the step � -+ � + 1 in the con struction of an elementary continuous chain so that the next structure is an enlargement of the previous one or even a saturated structure for a suitable cardinal - for instance adequate or good ultrapowers. As for the limit step, there is a simple universal construction.
Lemma 4.2.20. H 'Y is a limit ordinal or oo then any elementary contin uous chain of €-structures of length 'Y can be extended to an elementary continuous chain of length 'Y U { 'Y} ( = 'Y + 1 in the case when 'Y E Ord). Proof. Consider a elementary continuous chain which consists of structures ( � ; E� ) ( � < 'Y) and elementary embeddings e71� : V71 -+ � ( TJ � � < 'Y). Define V-y to be the collection of all pairs of the form (� + 1, x) , where � < 'Y and x E �+ 1 ' ran e� ,� + 1 J along with all pairs of the form (0, x) , x E V0 • Define (e, x) E-y (TJ', y) (�' , TJ1 being 0 or successor ordinals) iff e�' < (x) E{ e71, < (Y ) where ( = max{e, TJ' }. To define embeddings e�-y, � < 7, suppose that � < 'Y and x E � . There is a least ordinal ( � � such that x = e ZFC19 proves 4JvVK . Yet 4Jv is an axiom of ZFC19 . (iii) By definition, VK is an elementary submodel of V19 with respect to all En formulas, while V19 is a model of ZC plus En-Collection . In this case the proof of Theorem 1.5.4(ii) works.
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{iv) With "' playing the role of -8 , ZFC-D is satisfied by (ii), thus it remains to prove Minimality w. r. t. "' Suppose towards the contrary that K-1 < "' and VK' is an elementary submodel of VK with respect to all En( K) formulas {with parameters in VK' ). Note that n{K- ) ;:::: n{-D) by (iii). Thus VK' is an elementary submodel of ViJ with respect to all En(iJ) formulas, in contradic0 tion to the choice of "' . 0 Definition 4.5.2. ZFC-o+ is ZFC-D plus "V = L " plus Minimality. Corollary 4.5.3. The theory ZFc-o+ is equiconsistent with ZFC-D and hence with ZFC. Proof. It is known that ZFC + "V = L " is equiconsistent with ZFC. It fol lows by 1.5.17 that ZFC-D + "V = L " is equiconsistent with ZFC. It remains to apply Lemma 4.5.1{iv). {Note that the interpretation by "' does not change the set universe, thus does not violate "V = L " . ) 0 Blanket agreement 4.5.4. We argue in BST-D+ in the remainder of this section unless explicitly specified otherwise. 0 ·
4.5b The source of counterexamples An important consequence of Minimality is the existence of a sequence of sets yn E ViJ , "almost'' definable, but not really definable in ViJ . Let { 1/Jk {v)} kE N be a recursive enumeration of all parameter-free €-formulas with the only free variable v . Let rt'k (v) say that either v is the only set satisfying 1/Jk (v), or v = 0 1\ ..., 3 ! x 1/Jk (x), or, more formally, rt'k (v ) is (1/Jk (v) /\ 3 ! x 1/Jk (x)) V (v = 0 /\ • 3 ! x 'l/Jk(x)) . For each k E rN, let Yk be the unique set in ViJ satisfying ViJ F rt'k (Yk ), and a k be the least infinite ordinal < "' such that Yk � Vo:�c and O:k > O:k - l . Lemma 4.5.5. -D = supkEN a k . Proof. Let, on the contrary, -D > a = supkE N O:k . We claim that V0 is an elementary submodel of ViJ w. r. t. all En(iJ) formulas with parameters in V0 , in contradiction to Minimality. It suffices to prove that, for any m,
3 X E v{J (ViJ t= 4>(x)) => 3 k ;:::: m 3 X E VO:Ja (ViJ t= 4>(x)) , where 4>(x) is a En(iJ) formula with x as the only free variable and parame ters in Vo:m . We can assume that there is only one parameter Po (Po E V ) , so that 4> is 4>(Po, x). As ViJ models En{iJ)-Collection by Lemma 4.5.1, there is an ordinal v < "' such that V p E Vo:m (3 x 4>(p, x) => 3 x E V11 4>{p, x)) is true in ViJ . The least ordinal v of this kind is definable in ViJ, and hence v is equal to O:k for some k. We have then Vp E Vo: .... (3 x 4>(p, x) => 3 x E Vo:,. 4>{p, x )) a: ....
in ViJ, as required.
0
4.5 Non-reducibility of 1ST
161
Lemma 4.5 .6. The sequence { Yk}kerN is not definable in ViJ by an € formula (with parameters in v{J ). Proof. Let, on the contrary, a0 E VfJ , 4>(a, k, x) be an E-formula, and (V19 t= 4>(ao, k, x)) x = Yk
for all k E rN and x E V19 .
Then { ao} x rN also belongs to ViJ . Therefore, by Lemma 4.5.5, there is m E rN such that ao E Vo:m and moreover, all pairs (a0 , k), k E rN, belong to Vo:m . For any p = (a, k) E Vo:m , if there is a unique set x E Vt'J satisfying V{J F 4J(a, k, x), then this X is denoted by x(p); otherwise we put x(p) = 0. In particular every Yk belongs to the set {x(p) : p E Vo:m } . Note that the set Z = {p E Vo: m : p ¢ x(p)} belongs to ViJ and is €-definable in VfJ , and hence Z is equal to a set Yk , k E rN, therefore, equal to x(po), where Po = (ao, k) E Vo:m . This leads to a contradiction by the 0 diagonal argument: Po E Z Po ¢ x(po) = Z .
4.5c The ultrafilter Our priority in the construction of a standard core extension of ViJ will be to ensure that the map k � yk does not penetrate into the extension. Lemma 4.5.6 suggests the method: since no function definable in VfJ can real ize such a map, we have to define the extension in a form essentially definable in VfJ . To achieve this goal, maps f : V{J f in --)- VfJ definable in ViJ with parameters will be taken to form a quotient power. Accordingly we employ a VfJ-adequate ultrafilter with a very special property: the corresponding quantifier preserves the E-definability in ViJ . Now let us consider details. For any transitive set V, we use Def(V) to denote the set of all sets X � V, which are €-definable in V with parameters. More exactly, a set X � V belongs to Def (V) iff there exists an E-formula cp with parameters in V and a single free variable x, such that
X = {x E V : V t= cp(x)} = {x E V : cp(x) is true in V} . Let J = VfJ f in . Recall that, for an ultrafilter U � &(J), Ui P(i, x) means that the set {i E J : (i, x) E P} belongs to U .
Theorem 4.5.7. There is a VfJ -adequate ultrafilter U � &(J) satisfying the following: if P � JxVt'J , P E Def (ViJ ), then the set {z E VfJ : Ui P (i, z)} also belongs to Def (ViJ ) . Proof. One of the most important consequences of the axiom of con structibility "V = L " is the existence of a well-ordering 0, ( * ) x0 = G {Vp ( o) ' {x,. : 'Y < a}), where J.L(a) is the least ordinal J.L such that VJL � {x,. : 'Y < a} . (ii) Prove that this is a legitimate definition in the sense that ZFGT proves that for any A E Ord there is a set of the form { Xo} o < >. such that xo = 0 and ( * ) holds for all 0 < a < A . (Hint. First prove that G t X is a set for any set X - this is because G {x ) E x for any x, thus the result required needs only Separation in the language with G. Then note that to define {xo}o < >. we need only G t A, and hence this is a ZFC construction with G t A as a parameter.)
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4 Metamathematics of internal theories
{iii) Prove in ZFGT that a � x0 is a bijection Ord ont� V. Prove that the relation x (x, V1 , ... , Vn) is a formula of £e ,G in which all variables Vi occur only through expressions of the form v E Vi (that is, to the right of E ). Then ZFGT proves that for any formally Le,G definable classes X1 , ... , Xn the class X = {x : 4>{x, X1 , ... , Xn)} is a formally Le,G-definable class, too.
Proof. Argue by induction on the number of symbols in 4>, with (i) of Propo sition 4.6.8 used for elementary formulas 4> and (ii), (iii) for the inductive steps for ..., , A, 3 . We leave the proof as an easy exercise for the reader. 0 4.6c A nonstandard theory extending 1ST Our goal is to define a standard core interpretation of 1ST, even of a some what stronger nonstandard theory, in ZFGT. To introduce the stronger the ory, define Trutho T, a modified truth predicate, as the conjunction of the following st-E-formulas (compare with Definition 3.5.2 !) with free variables T and G : {1) T � ClForJilG , G is a function, S � dom G and G{x) E x for all x E S ; {2) ystp yst q ((rp = q, E T p = q) A (rp E q, E T p E q)) ; (3) for any standard cp, 1/J : rep A 1/J, E T => cp E T A 1/J E T, and r..., {cp A 'l/J), E T => r..., cp, E T V r..., 'l/J, E T. (4) for any standard cp : r..., cp, E T ==> cp ¢ T and
r..., ..., cp, E T => cp E T; {5) for any standard cp(vi) : r3 vi cp(vi), E T => 3st x {rcp(x), E T) and r..., 3 vi cp(vi), E T => yst x (r..., cp(x), E T) ; {6) ystp yst q (r G {p) = q, E T G {p) = q) . Accordingly, sets T satisfying Trutha T are called truth sets for (S ; G) . 1 9 A formula cp E ClForJilG is formally true {f. true) in ( S ; G) if there is a truth set T for (S ; G) containing cp . A formula cp is formally false (f. false) in (S ; G) iff ..., cp is f. true. Thus "cp is f. true {false) in S" are st-E-formulas with cp as the unique free variable. Similarly to Theorem 3.5.4{ii), no standard cp E ClForJilG can be both f. true and f. false in (S ; G) . Definition 4.6.10. ISTGT is 1ST plus the following axiom: (t) There is a function G such that S � dom G, G(x) E x n S for any standard x ¥:. 0, and any standard cp E ClForJilG is either f. true or f. false in ( S ; G) . In ISTGT, if G satisfies (t) then let To be the collection {not necessarily a set) of all standard formulas cp E ClForJilG which are f. true in ( S ; G) . G f S 0 is intended to interpret the formal symbol G of the language £e ,G · 19
Partial truth sets,
as
in § 3.5a, see footnote 23 on page 119.
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4 Metamathematics of internal theories
Exercise 4.6.11 (ISTGT). Prove, using (t), that if G is as indicated then 0 the structure ( S ; E, G t S, T a) interprets ZFGT. Thus ZFGT is interpretable in ISTGT, in a sense slightly different from § 1.5d as now the interpretation depends on a parameter G. Note that the basic universe of the interpretation is the class S of all standard sets. The result of 4.6.11 is the easier part of the following theorem:
Theorem 4.6.12. The theories ZFGT, ISTGT are interpretable in each other, in particular, the interpretation of ZFGT in ISTGT is given in 4.6.11 and it has S as the set universe, while ISTGT is standard core interpretable in ZFGT by Theorem 4.6.19 below. Remark 4.6.13. It takes some effort to derive, or even properly formulate, a reasonable conservativity result from Theorem 4.6.12. We conjecture that a sentence 4> of £e,G,T is a theorem of ZFGT iff ISTGT proves ( e , S, V G (G satisfies 4.6.10{t) => 4J s ; G t Ta) ) .
0
4.6d The ultrafilter To prove the nontrivial part of Theorem 4.6.12 we define an interpretation of the nonstandard theory ISTG T in ZFGT.
We argue in ZFGT in this subsection.
As usual, V = U� eord V� is the set universe of ZFGT. Put J = yfin = { x : x is finite} (a proper class, of course). Let !l) consist of all formally £e,G-definable classes X � J. Our first goal is to define, in ZFGT, a V-adequate ultrafilter U � !l) preserving the formal £e,G-definabi1ity as a quantifier. Fix a recursive enumeration {Xn(v) h { 1/J) indicate the coded formula 3 x l · ··3 xm ('l/J = r19(x 1 , ... , xm), A19 (xl , ... , xm)); it belongs to Forma . Thus T' = {1/J E Forma : r4>(1/J), E T}, so that by definition T' is formally .Ce,a-definable, and hence so is T by the above. It follows (see the construction of g and *G above) that there is a map r E § such that the element 'T = [r] E *V satisfies T { cp : � *E 'T} . On the other hand, one easily proves that T satisfies in V conditions {1) - (6) of § 4.6c with yst changed to V, G changed to G, and S � dom G dropped. It follows, since * is the natural embedding, that 'T satisfies Trutha 'T in *v. Finally, by definition, one of cp, ..., cp belongs to T, hence, one of the coded formulas �, ..., � *E-belongs to 'T . =
0 (Theorems 4.6.19 and 4.6. 12) 0 {The interpretability of 1ST in Theorem 4.1. 10)
4.6f Extendibility of standard models We argue in ZFC. Recall the notion of extendibility {Definition 4.1.6).
Corollary 4.6.21 ( ZFC ) . Any transitive set M such that, for some G : M � M and T � M, (M ; E, G, T) is a model of ZFGT, is 1ST-extendible. Proof. Apply Theorem 4.6.19 in (M ; E, G, T) .
0
Unlike BST-extendibility, it is not the case that any countable transitive model of ZFC is 1ST-extendible. Indeed, if there exist transitive models of ZFC then there is a unique minimal transitive model M of ZFC : it has the form M = Lp , where J.t is the least ordinal such that Lp (the set of all Godel constructible sets that appear at a level earlier than J.t in the construction of L, the class of all constructible sets) is a model of ZFC .
Exercise 4.6.22. Prove that J.t and M are countable, the axiom of con structibility "V = L " holds in M, and any x E M is €-definable in M, that is there is an E-formula cp( · ) with one free variable such that x is the only element of M satisfying M f= cp(x) . Hint. Show that the set M' of all x E M which are €-definable in M coincides with M. It follows from "V = L" in M that there is a well-ordering = {p E E : e(st p)} and still by definition e{st p) is -st p, and this is equivalent to 3s t x {p = ex), as required. The proof for o<e> is equally simple. The equivalences X = y ¢::=:} ex � ey and X E y ex ee f3.y for all x, y E 0 immediately follow from Lemma 5.2.4. We conclude that the map x � ex is an internal core embedding of ( 0 ; E, st) in e, in this case actually a st-E-isomorphism of ( 0 ; E, st) onto ( O ( e} ; eE , -st) .
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5 Definable external sets and metamathematics of HST
(iii) This follows from (ii) and Proposition 5.1.1 because in the case con sidered the formula 4>(x1 , , xn) is the same as 4>{x1 , . . . , xn) ( D ; E , s t ) while e{4>( exl , ... , exn) int ) is the same as 4>{ eX l , ... , exn) ( o<e> i -e , 't ) . (i) We have to prove e4> in BST for any axiom 4J of EEST . Extensiona l ity. Suppose V r E E {r eE p {=::} r eE q ) where p, q E E. In particular ex eE p {=::} ex ee q for any X. However ex ee p means X £ p by Lemma 5.2.4{i), and hence we have V x (x £ p {=::} x £ q), that is p "== q . Transitivity of D. It suffices to show that, for p, q E E, if p eE q then -:i.nt p. However p eE q by definition implies that p "== ex for some x £ q. Now p = ex and -:i.nt p follow from Lemma 5.2.4{ii). To prove Separation in e suppose that P E E and 4>{x) is a st-E-formula with parameters in E ; we have to find Q E E such that ..•
Vx (x eE Q (x eE P A e4>{x)) ) . By Theorem 3.2.16 it suffices to show that X = {x : X eE p A e4>{x)} is an "external set." The st-E-definability of X (with the parameter P and the parameters involved in 4>) is obvious, thus it remains to find a set Y (internal) with X � Y. Clearly Y = ran P is as required because Ep � Y. Union easily follows from Separation. To prove e{BSTint ) apply (iii). Parametrization holds by the construction of e . Indeed consider any p E E. Let q = ep, then q E E and p = Eq . Formally, q E E A p = Eq . Thus we have e((q E E A p = Eq) int ) by {iii). However the formula q E E A p = Eq is absolute for the internal universe: apply the Transitivity of D and e{BSTint ) . Thus finally e(q E E A p = Eq) as required. 0 0 (2° of Theorem 5.1.4)
5.2b Elementary external sets in external theories By definition, both in HST and in EEST, the class D of all internal sets (that is elements of standard sets) or, to be more exact, the structure ( D ; E, st) , satisfies BST {for HST by Theorem 3.1.8). In addition D is transitive by the axiom of Transitivity of D.
Definition 5.2. 7 {HST or EEST) . Define E, for any internal p as in Def inition 3.2.14, and E as in Definition 5.2.1. Define the formulas £, e= , eE , -st, �nt as in Definition 5.2.2. An elementary external set is any set of the form E, , p E E. [ = {E, : p E E} , the class of all elementary external sets. 0 Lemma 5.2.8 (HST or EEST ) . If p E E then E, is a set . Proof. Apply Separation in the st-E-language.
0
Thus differently from BST all "external sets" are true sets in HST and in EEST ! This enables us to change notation in external theories from "external
5.2 From internal to elementary external sets
187
sets" to elementary external sets as in Definition 5.2.7. Note also that the definitions of E and x E E, are absolute for 0 because 0 is a transitive class and an interpretation of BST both in HST and in EEST. In other words, it does not matter whether we define E or E, for some p E E in 0 or in the whole external set universe of HST or EEST. This allows us to use all related theorems in §§ 3.2f, 5.2a in HST and EEST.
Exercise 5.2.9 (EEST ) . Prove using the Parametrization axiom that [ con 0 tains all sets ! Why is this not the case in HST? Thus elementary external sets in external theories are the same as "ex ternal sets" in BST. The word elementary refers to the fact that, first, all sets in [ are subsets of 0, and second, all of them admit a direct coding by means of internal sets. Hence [ is arguably the family of simplest possible external sets (some of them are internal, of course) - this explains why we call sets in [ elementary external. The next theorem shows that [ coincides with another family of sets considered in § 1.4a.
Theorem 5.2.10 (HST ) . The following classes coincide: A28 , [, and the class of all "external sets '' in the sense of the universe 0, that is, the class of all sets X � 0 st-E-definable in 0 (with parameters in D). Proof. That "external sets" = [ follows from Theorem 3.2.16. 3 To show that A28 � [, consider a A28 set X = U a eu n b e V Xab , where U, V E WIF and all sets Xab are internal. By Corollary 1.3.13{ii) {Extension) there exists an internal function p defined on *U x *V so that p(*a, *b) = Xab for all a E U, b E B. Now we have p E E and X = E, . To prove the converse, let X = E, , where domp = *U x *V; U, V being well-founded sets. We put Xab = p(*a, *b) for all a E U and b E V. Then X = U a e u n b e V Xab· To get a n U-presentation of X, take a standard set S such that X � S and consider the complement X' = S ' X . 0 In the remainder, it will be more important that the class [ contains all sets st-E-definable in 0, while the presentation implied by A28 will have some technical applications.
Theorem 5.2. 11 ( HST or EEST) . [ is a transitive class containing all internal (hence all standard) sets and satisfying EEST. In particular the structure e = ( [ ; E, st) is an interpretation of EEST in HST . Proof. It follows from Lemma 5.2.4 that the map p � Ep is a reduction of the invariant structure e = (E; ee , -st ; e= ) to the structure e = ( [ ; E, st) with true equality in the sense of Definition 1.5.9, and hence we have HST : 4J ( Ep l , . .. , Epn ) IE e4J{p l , ... ,pn) ( e4J{p l , ... , pn)) int ( *) EEST : 4J ( Ep p ... , Epn ) e4J{p 1 , ... ,pn) ( e4J (p l , ... , pn)) int
}
3 Give a precise formulation of this fact, as in Theorem 3.2.16(i).
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5 Definable external sets and metamathematics of HST
for any st-E-formula 4>(xl , .. . , Xn ) and any Pl , ... , Pn E E by Prop osi tion 1.5.10. The rightmost equivalence in both lines holds since the domain E of the structure e is a subclass of 0 anyway. The superscript [ is omitted in the first term of the EEST line because [ contains all sets in EEST . 0 Now the theorem immediately follows from Theorem 5.2.6(i).
Exercise 5.2.12. Prove the equivalences (* ) directly by induction on the complexity of 4> using Lemma 5.2.4. Also prove the following in HST : (1) [ contains all sets X � 0 of standard size. (2) N and IHIF (both non-internal sets; see § 1.2e on IHIF) belong to [. (3) 0 � [ � IH . (4) The [- "power set" &(X) n [ is not a set for any infinite internal set X . Hints. (1) By Theorem 1.3.12 there is a standard set S and an internal function f defined on S such that X = {f(y) : y E S n S}. Then X is st-E-definable in 0 with parameters /, S, and hence X E [. (2) Both sets are standard size subsets of 0 . (3) The axiom of Transitivity of 0 proves X � 0 for any X E [. Codes of the form ex witness that 0 � [. Finally, N E [ ' 0 while { N } ¢ [. (4) Apply Theorem 1.3.9. 0 Proof of part
6°
of Corollary 5.1.5
Given a st-E-sentence 4>, let cp be the sentence e4>. Then EEST proves 4> cpint by the ERST-equivalence in the proof of Theorem 5.2.11. To get an €-sentence 'l/J satisfying 4> 1/J s t in EEST apply the reducibility of BST by Theorem 4.1 . 10(ii) to cp, together with the fact that 0 interprets BST in EEST because the latter includes BSTint .
5.2c Some basic theorems of EEST Here several important theorems of EEST are presented. All of them cap italize on basic theorems of BST (§§ 3.2d, 3.2e) . In general the axiom of Parametrization effectively reduces properties of the EEST set universe [ to its internal universe 0 which, as we know, satisfies BST. Note that by Theorem 5.2.11 formal deduction in EEST can be employed to study elementary external sets in HST.
Lemma 5.2.13 (EEST) . Every set C is a subset of a standard set. Proof. By Parametrization , C = E, , where p E E, thus, p E 0 and C � Y = U ran p. Yet the set Y is internal (define U ran p in 0 and prove, using Transitivity of 0 , that this is Y). Now apply 3.1.7(iii) in D . 0 Theorem 5.2.14 (EEST) . Let 4>(x, y) be a st-E-formula with arbitrary sets as parameters. For any set X there exist standard sets S, Y and an internal function F such that the following holds:
5.2 From internal to elementary external sets
189
Standardization: S n S =
XnS; Collection: V x E X (3int y 4i(x, y) ==> 3 y E Y 4i(x, y) ) ; Standard Size Choice: ys t x E X (3int y 4i(x, y) ===? 4i(x, F(x))) .
Proof. B y Parametrization any parameter in 4i has the form Eq ; let, for brevity, 4i be 4i(x, y, Eq), q E E. Let !li(x, y, q) be the formula e4i( ex, ey, q). Then 4i(x, y, Eq) iff !P(x, y, q) holds in 0 (by (* ) of Theorem 5.2.6) . Now to obtain Y apply Theorem 3.2.8 {BST Collection) in 0 to the formula !P. To obtain F and S use resp. the BST I nner S. S. Choice (Theorem 3.2. 11) and 0 the BST axiom of Inner Standardization the same way. Definition 5.2.15. In EEST a set-like collection is any collection of the 0 form {E, : p E P } , where P � E is a set. This informal definition is convenient to meaningfully consider in EEST collections that are not sets. For instance, if at least one of sets x, y is not internal then { x , y } is not a set (see Exercise 5.2. 17(1) below) but clearly a set-like collection. The following theorem ensures a rather good behaviour of these objects.
Theorem 5.2.16 ( EEST ) . Let rp(x) and 4i(x, y) be st-E-formulas with arbitrary sets as parameters. For any set-like collection X there exist set-like collections X' , Y such that Separation: Vx E X (x E X' ¢::=:> rp(x)) ; Collection: V x E X ( 3 y 4i(x, y) ===? 3 y E Y 4i(x, y)) . Proof. Let X = {E, : p E P } , where P � E is a set. Then, by the Separation axiom, P' = {p E P : rp(E,)} is a set. Yet X' = {E, : p E P'} . To prove Collection apply Collection of Theorem 5.2. 14 to the formula !li(x, p) saying p E E A 4i(x, E,). 0 Exercise 5.2. 17. Rewrite the statement of Theorem 5.2.16 in the ordinary language of EEST, in terms of E-codes. Also, prove the following in EEST : (1) If x E y then x is necessarily internal (by Lemma 5.2. 13, y � S for a standard set S), but y may be non-internal. (2) Any set-like collection that consists only of internal sets is a set. (3) (Difficult !) If X is an infinite standard set then & (X ) is not a set-like collection. (Hint. See Hrbacek paradox, Theorem 1.3.9.) 0 5.2d Standard size, natural numbers, finiteness in
EEST
We accept, for EEST, the same definition of sets of standard size (Defini tion 1.1. 12) as in HST. Note that not all of Theorem 1.3.1 remains true in EEST because the well-founded universe is too small, see below.
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5 Definable external sets and metamathematics of HST
Theorem 5.2.18 (EEST) . The Saturation axiom, as in § 1. 1f, holds. Proof. Let !C � 0 be a n-closed set of standard size consisting of non empty sets; we prove that n !C i:. 0. By definition there is a set S � S and a function f with S � dom f such that !C = {f(s) : s E S} . We can assume that f is internal, by Theorem 5.2.14. Now apply Theorem 3.1.23 in 0 . 0 Exercise 5.2.19 (EEST). Prove that the Dependent Choice axiom, as in § 1. 1f, holds. Hint: apply, in 0, Theorem 3.2.11 (Inner Dependent Choice). 0 Exercise 5.2.20 (EEST). Prove that the set rN = w of all standard 0natural numbers is the largest ordinal. (Hint. To prove that rN is well-ordered argue as in the proof of Lemma 3.1.18(i). Then note that in EEST a non 0 internal set like rN cannot be a member of a set.) In EEST, by ordinals we still mean transitive sets well-ordered by E . Thus by 5.2.20 the class Ord of all ordinals is too miserable, just w U { w} w + 1. Fortunately there is a good replacement. Let SOrd be the class of all S ordinals, i. e. standard sets that are ordinals in the sense of S. =
Lemma 5.2.21 (EEST) . The class SOrd is well-ordered by E. Moreover for any set X � SOrd there exists a least S-ordinal a ¢ X. This ordinal will be denoted by a = sup5 X . Proof. B y Theorem 5.2.14 there is a standard set Y � SOrd such that X n SOrd = Y n SOrd. As S satisfies ZFC there exists a least S-ordinal 0 a ¢ Y. By the choice of Y this a is as required. Elements of the set rN are called natural numbers. A finite set is a set equinumerous to { 1, 2, . . . , n } = { k : 1 :$ k :$ n } , where n E rN . Thus, natural numbers in EEST are S-natural numbers, i. e. standard sets n such that it is true in S (or, equivalently, in 0 ) that n is a natural number. Similarly to § 1.2e, define IHIF = Un e rN & n (0) (all hereditarily finite sets). The next exercise shows that the domain of *-methods in EEST is restricted to elements and subsets of IHIF.
Exercise 5.2.22 (EEST). Prove the following: (1) IHIF = "'IHIFnS, where "'IHIF is the internal set of all internal sets hereditarily finite in 0, hence, IHIF is a set; (2) a set x is well-founded iff x � IHIF, i. e. , WIF = & (IHIF) (compare with Exercise 3.2.20), in particular, rN E WIF and rN � WIF; (3) *x E 0 can be defined, as in 1.1.6, for any x � IHIF, and we have *x = x for any x E IHIF but x � *x for any infinite x � IHIF; (4) 1R � WIF, therefore, *r E 0 is defined for any real r, but IR itself is not a set in EEST . o
5.3 Assembling of external sets in HST
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5.3 Assembling of external sets in HST
Elementary external sets (the class [ = A�8 ) are characterized, within all external sets, by two properties: 1st, their definability, and 2nd, the fact that they contain only internal elements, that is, they are sets of the 1st von Neumann level over 0 (see § 1.5b). Obviously there are many definable external sets of higher levels. For instance, any monad of a standard real (§ 2.1a) is an elementary external (and non-internal) set, thus the set of all monads is a definable external set of second level over D. Sets of higher levels can also be defined. There is a universal method to present this multitude of external sets. Recall that in set theories containing Regularity, like ZFC, the construction of an arbitrary set x can be presented as a well-founded tree T, with the empty set assigned to every endpoint of T, such that at any preceding point we assemble all sets already assigned to its immediate successors, and x, the given set, comes out at the root. In HST, the axiom of Regularity over 0 allows to define sets in a similar manner, but endpoints of trees have to be assigned, or "decorated" with arbitrary internal sets, not necessarily the empty set. Graph theory calls such a construction a decoration of a tree. This section presents the construction itself. It will have two major appli cations: an interpretation of HST in EEST in Section 5.4, and a class ll.. [ D] of all sets obtained by assembling beginning with internal sets in Section 5.5. (Sets in ll.. [ D] will be called sets constructible from internal sets.) In those applications, a particular form of the construction will be used, such that the trees and assigments to endpoints belong to [, in order to obtain all external sets st-E-definable in the broadest sense.
5.3a Well-founded trees Let Seq denote the class of all sequences (a1 , ... , an) (of arbitrary sets ai , but mostly only internal ai will be considered) of finite length. For t E Seq and every set a, t Aa is the sequence in Seq obtained by adjoining a as the rightmost additional term to t. The notation a At is understood correspondingly. Generally, s At E Seq is the concatenation of two sequences s , t E Seq. The formula t' � t means that the sequence t E Seq extends t' e Seq (perhaps t' = t), while t' C t will mean that t is a proper extension of t' (so that t' ¥:. t). (a) is a sequence with the only term a . - A tree is a nonempty set T � Seq such that, for any pair of sequences t', t E Seq satisfying t' � t, we have t E T ==> t' E T. Note that every tree contains A, the empty sequence. - Define Max T to be the set of all �-maximal elements r E T. - H t E T then let Succr (t) = {a : t Aa E T} . - Define Min T = Succr (A) = {a : (a) E T}; then Min T = 0 iff T = {A} .
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- A tree T is well-founded ( wf tree, in brief) if every nonempty set T' � T contains an element �-maximal in T' . Thus, a tree T is well-founded if the inverse relation -< ( t -< t' iff t' C t) is a well-founded relation on T in the sense of Definition 1.1.4. Therefore, the next definition is a legitimate definition by well-founded induction (see Remark 1.1.7), as so is any definition of the following kind: a function f is defined on a given wf tree T so that each value f (t) depends only on values f(t Aa) , where t Aa E T.
Definition 5.3.1. Let T be a wf tree. Define an ordinal ! t i T (the rank of t in T) for each t E T so that l t i T = supt " aET l t Aa i T · In particular, I t i T = 0 for t E Max T (since sup 0 = 0). 0 Put ITI = IAIT (the height of T) . Exercise 5.3.2. Suppose that T is a wf tree. Prove that if a set X � T satisfies Max T � X and is inductive in T (that is t E X whenever t E T is such that t Aa belongs to X for all a E SuccT (t) ) then X = T. Hint. Assume otherwise and consider a �-maximal element t E T ' X. ) 0 Prove that for any t E T then there is t' E Max T with t � t' . 5.3b Coding of the assembling construction The following definition formalizes the idea of assembling construction.
Definition 5.3.3. 4 An A-code (or: assembling code) is any function x : D -+ 0 defined on a set D � Seq consisting of pairwise �-incomparable sequences, such that Tx = {t E Seq : 3 t' E dom x : t � t'} is a wf tree. (Note that then Max Tx = D = dom x, therefore x is a map Max Tx -+ 0 .) In this case, a function Fx (·) can be defined on Tx, by the same kind of well-founded induction as above: 1) if t E Max Tx then Fx(t) = x (t) ; 2) if t ¢ Max Tx then Fx(t) = {Fx(t Aa) : t Aa E T} . 0 We define Ax = Fx(A) (the set coded by x). It is, perhaps, more natural to define an A-code to consist of a well founded tree T � Seq and a function x : Max T � D. Then the function Fx ( · ) defined on T in accordance with 1) and 2) is a decomtion of T (relative to x ) in the notational system of graph theory (see, e. g., Devlin [Dev 98]). However this would lead to a certain technical inconvenience. Indeed, we shall be mainly interested in those A-codes which belong to [. As the class [ is, generally speaking, not closed under pairing, a pair consisting of a wf tree T and a function x is not necessarily a member of [ even if both T and 4 In this definition, A in all shapes refers to "assembling" .
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x separately belong to [. This technical nuisance can be fixed by different means; our solution is based on the fact that T, a wf tree, is obviously a function of any suitable x : as above just take the transitive C -closure Tx of dom x downwards. 5.3c Examples of codes We introduce here several useful types of A-codes. To begin with, consider codes of intermediate sets Fx ( t) .
Exercise 5.3.4. Suppose that x is an A-code. For any element t E T = Tx, we put Tit = {s : t As E T} . Further, define x lt (s) = x(t As) for any s E Max Tit = {s : t As E Max T}. In particular, if a E Min T then (a) (a one-term sequence) belongs to T - thus we can define Tla = {s : a As E T} and x la (s) = x(a As) for any s E Max Tia = {s : a As E T} . Exercise: prove that x l t is an A-code and Ax l t = Fx (t) for any t E T = Tx, in particular, x l a is an A-code and Ax l a = Fx ((a)) for any a E Min T, moreover, if ITI � 1 (so that T f. { A } ) then Ax = { Ax l a : a E Min T} . 0 It occurs that any set x E IH is equal to Ax for a suitable A-code x. This will be another useful family of codes.
Definition 5.3.5. Let x be any set. Define an A-code &a; with x = A-x as follows. If x is internal let Tax = {A} and &a;(A) = x. In other words, in this case &a; = { (A, x) }. If x ¢ 0 then let Tax be the set of all finite sequences of the form t = (y0 , y1 , ... , Yn) , where n E N, Yi are arbitrary sets all of which except possibly Yn are non-internal, and x 3 Yo 3 Y1 3 ... 3 Yn, together with the empty sequence A. (Tax is a set, for instance, because any sequence t E Tax consists of sets which belong to the transitive closure of x . ) Then Tax � Seq is a tree and Max Tax consists of all sequences t = (yo, ... , Yn) with 0 Yn E D . Put Ba;(t) = Yn for any such t . Lemma 5.3.6. Tax � Seq is a wE tree, &a; is an A-code, and A-x = x. Moreover, Fax (t) = Yn and (Ba;) lt = fl.yn for any t = (yo, ... , Yn) E Tax . Proof. Suppose towards the contrary that Tax is not a wf tree. Then, by Dependent Choice, there exists an infinite sequence y 3 bo 3 b1 3 .. . of non-internal sets, clearly a contradiction to Regu l arity over 0 . Therefore, we can prove the first equality of the "moreover" statement by well-founded induction, on the base of Exercise 5.3.2, i. e., prove that it holds for all t E Max Tax , and also holds for any t E Tax provided it holds for all immediate successors t Aa E Taz . If t = (ao, ... , an) E Max Tax , so that an E 0, then by definition Fax (t) = �(t) = an. Suppose that t ¢ Max Taz . All immediate successors of t in Tax are of the form t A a = (ao, ... , an, a) , where a E an. If Fax (t Aa) = a for all a E an then
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F-x (t) = {F-x (t Aa) : a E an} = a n ,
as required. In particular, A-x = f-x (A) = {F-x ((a)) : a E x} = x .
0
Let us mention a related problem in passing by.
Problem 5.3. 7. Is it true that every set x is equal to Ax for an A-code x such that Tx � 0 - that is, the tree Tx consists only of (finite) sequences of internal terms ? A positive answer would follow from the hypothesis that every set is a functional image of an internal set. This hypothesis is consistent with HST (claim 8° of Theorem 5.5.4 below), but most likely not provable in HST . 0 Example 5.3.8. The codes ax will be especially important for sets x � D. If x is internal then, by definition, T-x = {A} and ax(A) = x . If x � 0 is non-internal then T-x = {A} U { (a) : a E x} (hence, Max Tax = {(a) : a E x} 0 and Min T-z = {a : a E x} = x), and �((a)) = a for any a E x . Now let us consider A-coding of well-founded sets. We leave it as an
exercise for the reader to show that if v E WIF then ax E WIF as well. In particular, any t = (a0 , , an) E T-x is a (finite) sequence of well-founded sets ai , which, except possibly an, are non-internal. Recall that •••
WIF n 0 = WIF n S = IHIF = { v E WIF : *v = v} is the (well-founded, non-internal) set of all hereditarily finite sets (Exer cise 1.2.17). thus the requirement of non-internality can be reformulated as follows: none of the (well-founded) sets ai, except possibly for an, belongs to IHIF. The next definition introduces an isomorphic copy of B.u, whose advan tage is that the associated wf tree consists of internal (moreover, standard) sequences.
Definition 5.3.9. Suppose that v E WIF and y = *v (a standard set). If v E IHIF, and hence y = v, then put c[y] = &.y. Suppose that v ¢ IHIF. Let T[y] be the set of all finite sequences of the form t = (bo, b l , ... , bn) , where n E N, v 3 bo 3 b1 3 . . . 3 bn, and bi are arbitrary standard sets (thus, bi = *ai for a well-founded set ai ) - with the restriction that all of them, except possibly for bn, do not belong to IHIF, together with the empty sequence A. Thus Max T[y] consists of all sequences t = (bo, ... , bn) E T[y) with bn E IHIF. Put c[y] (t) = bn (then = *bn as x = *x 0 for x E IHIF) for any such t . Exercise 5.3.10. Prove, using Lemma 5.3.6, that, in both cases, T[y] � Seqn S is a wf tree, c[y) is an A-code, and Ac[y] = v. Finally, if v ¢ IHIF then Fc[y] (t) = an and c[y) l t = c[an] for any t = (*ao , ... , *an) E T[y] . 0
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5.3d Regular codes Any internal set x admits not only the "natural" code ax, but many other codes, for instance, a code that assembles x in one step from its elements. To inhibit such a non-uniqueness, consider the following special class of A-codes that produce internal sets only through "natural" codes of the form ax .
Definition 5.3.11. An A-code x is regular if for each t E T = Tx satisfying 0 !tiT = 1 the set Fx (t) = {x(t Aa) : t Aa E Max T} is not internal. Thus regularity requires that internal sets do not appear at the first assem bling level. The next lemma shows that this requirement is sufficient to forbid internal sets to appear at all higher levels. Let Dix = { t E Tx : Fx (t) E D}, the domain of internality. Note that any code x with Tx = {A} is regular, and Dix = Tx = {A} .
Lemma 5.3.12. An A-code x is regular iff Dix = Max Tx . Proof. If Dix = Max Tx then x is regular by definition. To prove the converse suppose that x is regular. Since Max Tx � Dix for any A-code, it remains to check the opposite inclusion. Let X = { t E Tx : t ¢ Dix V t E Max Tx} · We have to show that X = Tx . According to Exercise 5.3.2, it suffices to prove t E X, assuming that t E Tx and every extension t Aa E Tx belongs to X. Let, on the contrary, t ¢ X. Then t E Tx ' Max Tx and x = f {t) E D. We have 1t 1 Tx � 2 because of regularity, hence there is t A a E Tx such that lt AaiTx � 1. Thus t Aa ¢ Max Tx. Yet t Aa E X, and hence t Aa ¢ Dix and y = Fx (t Aa) ¢ D. However y E x , a contradiction to Transitivity of D . 0 x
Exercise 5.3.13. Prove that if an A-code x is regular then irk Fx(t) = I tiT 0 for all t E T = Tx. (Recall that irk is the rank over D, see § 1.5b.) Yet our coding potential does not really suffer, because any A -code x can be reduced to a regular A-code xR such that Ax = AxR · To define xR note that the set D of all �-minimal elements of Dix is obviously pairwise �-incomparable, that is s � t for all s f:. t in D . Put xR(t) = fx { t) for t E D ; note that xR(t) E D because D � Dix .
Exercise 5.3.14. Prove that then xR is a regular A-code satisfying Ax = = (Tx ' Dix} U D, Max TxR = D. In addition, prove the following: (i} All codes ax and c [y) , y E S (Definitions 5.3.5, 5.3.9} are regular. (Hint: regarding c [y) , apply the result of 1.3.8{2} that a standard size set of internal sets is internal iff it is finite.) (ii) An A-code x with ITx l � 1 is regular iff all codes xl a , a E Min Tx {Ex ercise 5.3.4} are regular and either ITx l � 2 or ITx l = 1 and Ax ¢ D . 0 (iii) If x is a regular A -code and Ax = x E D then x = ax . AxR , TxR
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5.4 :From elementary external to all external sets
The main goal of this section is to prove 3° of Theorem 5.1.4. In the course of the proof we define an interpretation of HST in EEST based on the as sembling construction outlined in Section 5.3. Blanket agreement 5. 4. 1 .
In the arguments below [ will denote: - either the set universe of EEST and then we study it by means of EEST {the EEST case) ; - or the class of all elementary external sets in HST and then we study it by means of HST {the HST case). Note that in the HST case [ still satisfies EEST by Theorem 5.2.11. We shall make clear distinctions whenever it is necessary to avoid ambiguity. As usual S � 0 � [ are classes of resp. standard and internal sets in [. 0 Thus S = { x E [ : st x } and 0 = { x E [ : int x } . -
-
The HST case will lead to the class ll.. [ O) of all sets constructible from internal sets in Section 5.5 while the EEST case is directly connected with the interpretation of HST in EEST defined below in this section. The domain of the interpretation will consist of all regular A-codes x E [. The intended meaning of the basic relations ae , 11::::: , 8st is connected with the coded sets Ax, for instance x 8E y iff Ax E Ay. The main difficulty here is that the sets Ax themselves generally speaking do not belong to [, and hence we cannot explicitly appeal to any relation between them. To solve the problem we shall find adequate definitions of basic relations within [. Proofs of items 4 ° and 5° of Corollary 5. 1.5 follow in § 5.4f. This section ends with a continuation of our discussion of external sets in BST which began in § 3.2f.
5.4a The domain of the interpretation First of all let us have another look at different notions introduced in Sec tion 5.3 from the point of view of [ as the principal domain. In [ only internal sets can be elements of other sets, and hence Seq consists of finite internal sequences of internal sets. The method of definition by well-founded induction on a wf tree has to be somewhat changed in [. Indeed it follows from 5.2.20 that the class Ord of all ordinals is equal to w U { w } in EEST, and hence is too small to support transfinite induction of any bigger length. Thus the rank function l tiT : T -+ Ord generally does not exist in [ for a wf tree T � Seq. Yet Lemma 5.2.21 provides us with an equivalent substitution in the class of S-ordinals. Say that T E [, T � Seq is a [-w/ tree if every nonempty set T' E [, ' T � T contains an element �-maximal in T' . This is the same as just being wf in the EEST case (see 5.4.1 on the cases). Let us show that this is also the same in the HST case.
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Definition 5.4.2. Let T E [ be a wf tree. Define an S-ordinal I t I;. for each t E T so that ltiT = sups t " a e T lt AaiT for any t E T {the least S-ordinal strictly bigger than all S-ordinals lt AaiT, t Aa E T). In particular, l t i T = 0 for t E Max T. Put I T I * = IAIT · 0 Lemma 5.4.3. {i) For any [-wf tree T E [ there is a unique map t � I t i T from T to SOrd that belongs to [ and satisfies I t I;. = sups t " a e T I t A a IT for all t E T ; (ii) {HST ) IT I ; = *{I Ti t ) for any [-wf tree T E [ ; (iii) (HST ) any tree T E [ is wE iff it is [-wf. Proof. {i) Let, for any t E T, a t-function be any map I E [ defined on the set { t' E T : t � t'} and satisfying I tiT = sups t " a ET I t A a IT on its domain. It suffices to prove that for any t E T there is a unique t-function It · Note that "being a t-function" and the existence of a t-function are st-E formulas relativized to [. It follows that if I i:. g are two t-functions fot some t E T then the set Xtfg = { t' E T : t � t' A l(t') i:. g(t')} is st-E-definable in [. We conclude that Xtfg E [ because [ satisfies EEST by Theorem 5.2.11. If I i:. g then Xtfg i:. 0, and hence Xtfg contains a �-maximal element t' (inteed T is [-wf). Thus l(t') i:. g (t'), but l(r) = g (r) for any T E T with t' C r, easily leading to contradiction. This proves the uniqueness of ft. To prove the existence suppose towards the contrary that X = { t E T : there is no t-function} ¥:. 0. Still X belongs to [ and hence it contains a �-maximal element t. In other words a unique It " a E [ does exist for any a E SuccT(t), but It does not exist. Define I = U a esucc T ( t} lt " a · In addition, let l(t) be the least S-ordinal bigger than all S-ordinals lt " a (t Aa ), a E SuccT(t). (To see that this is well-defined use Lemma 5.2.21.) Clearly I is st-E-definable in [, and hence it belongs to [ by Theorem 5.2.11. It follows that I is a t-function, contradiction. (ii) A routine proof with the help of usual HST methods including * Transfer is left for the reader. (iii) As S-ordinals are isomorphic to the true ordinals (those in WIF ) in HST via *, and hence are well-ordered, the map t H- l tl;. proves that any 0 [-wf tree T E [ is wf in the universe of the HST universe as well. We keep the definition of A-code and Tx as in 5.3.3.
Definition 5.4.4 (EEST ) . A is the class of all A-codes x E [. A is the class of all regular A-codes in A, where the regularity means that { x (t Aa ) : t Aa E Tx } ¢ 0 whenever t E Tx satisfies ITI; = 1 { i. e. t ¢ Max Tx 0 but any t' E Tx with t C t' belongs to Max Tx ). Due to the restrictive character of the EEST set universe the functions fx{·) generally speaking do not exist for x E A, accordingly, Ax, generally speaking, cannot be defined as in Definition 5.3.3.
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Theorem 5.4.5 . The class A is st-E-definable in [ . Proof. In the EEST case (see 5.4. 1) the result is obvious. In the HST case we have to prove that the definition of A is absolute for [ in HST. The absoluteness easily follows from Lemma 5.4.3(iii), because all other elements of the definition of A (that is except for the well-foundedness of the tree) are 0 absolute by rather obvious reasons. Lemma 5.4.6. If x E A then Tx, MaxTx, Min Tx are sets in [ while x l t, x l a for all t E Tx, a E Min Tx are sets in [ and codes in A. In addition, 8Z for any z E [ and c[x] for any standard x are sets in [ and codes in A . Proof. Routine verification based o n some results of § 5.2c, most notably, Lemma 5.2.13 and EEST Separation. For instance, if x E A then for mally x is a function with D = dom x � Seq. On the other hand, by Lemma 5.2.13 there is a standard, hence, internal set P with x � P. Then dom P = {x : 3 y ((x, y) E P)} is still an internal set (define it in 0, which is an €-interpretation of ZFC ). It follows that D is a set by Separation. Now, still by Lemma 5.2. 13, D � T, where T is internal, and we can assume that T � Seq. Moreover, T' = {t' E Seq : 3 t E T (t' � t)} is an internal set, hence, Tx (a definable subset of T' ) is a set by Separation. If x is internal then by definition � = { (A, x)} E D. If x E [ ' 0 then � = { (( a) , a ) : a E x} (see Example 5.3.8). Choose an internal set S with x � S (Lemma 5.2.13). Then � is a subset of the internal set X = { (( a) , a) : a E S} st-E-definable in [, anf hence 8X E [ by Theorem 5.2.16. It follows that "'x E A for any x E [. As the regularity is obvious we have "'x E A . 0 We leave the rest of the lemma as an exercise for the reader. The class A will be the domain of the interpretation.
5.4b Basic relations between codes
We continue to argue under the assumptions of 5.,4.1. Suppose that x, y E [ are A-codes in A. In principle, to figure out
whether, say, Ax = Ay , we have to compute both sets and check whether they are equal - but this is impossible within [ because the coded sets do not necessarily belong to [. Yet there is a way to avoid the actual computation of coded sets, based on the following definition taken from graph theory.
Definition 5.4. 7. A map j : Tx x Ty -+ { 0, 1 } is a bisimulation for A-codes x and y if it satisfies the following requirements: 1* . If t E Max Tx and r E Max Ty then j(t, r) = 1 iff x(t) = y(r) . 2* . If t E Max Tx but r ¢ Max Ty, or conversely, t ¢ Max Tx but r E Max Ty, then j(t, r) = 0 . 3* . Suppose that t ¢ Max Tx and r ¢ Max Ty. Then j(r, t) = 1 iff
5.4 From elementary external to all external sets
(a) V r Ab E Ty 3 t Aa E Tx (j(t Aa, r Ab) = 1) , and {b) V t Aa E Tx 3 r Ab E Ty (j(t Aa, r Ab) = 1) .
199 0
Since we consider only well-founded trees, in HST for any two A-codes x, y E A there exists a bisimulation j : Tx x Ty � { 0, 1} defined so that j(t, r) = 1 whenever Fx (t) = Fy {r) and j(t, r) = 0 otherwise. (The require ment of regularity validates 2* ; in the non-regular case that would be more cumbersome.) Now, under the assumptions of 5.4.1, we prove
Lemma 5.4.8. For any two codes x, y E A there exists a unique bisimula tion j. This unique bisimulation belongs to [. It will be denoted by jxy . Proof. We argue as in the proof of Lemma 5.4.3(i). Let, for t E Tx and r E Ty , a (t, r)-function be any function j E [ defined on the set { (t', r') E Tx x Ty : t � t' A r � r' } and satisfying 1 * , 2*, 3* of Definition 5.4. 7 on this domain. Let P(t) say: "for any r E Ty , there is a unique (t, r)-function" . To prove P(A ), the desired result, it suffices to show that the set of all t E Tx with P(t) is inductive (see Exercise 5.3.2). Take any t E Tx, suppose P(t Aa) for all extensions t Aa E Tx (for instance, this holds for t E Max Tx ), and derive P(t). Let r E Ty. By the inductive hypothesis, for any t Aa E Tx and r Ab E Ty there exists a unique (t Aa, r A b)-function, say, iab E [. Moreover, it follows from the uniqueness that these functions are pairwise compatible on intersections of the domains, and hence the union j = Uab iab is a function. Since "to be a {t, r)-function" is a notion absolute for [, the formula which defines j witnesses that j is st-E-definable in [, therefore j E [ because [ satisfies EEST by Theorem 5.2.11. It remains to define, additionally, values j(t, r') and j(t', r) for all r' E Ty with r � r' and all t' E Tx with t � t' (in particular, j(t, r) ) applying Definition 5.4.7: the result belongs to [ still by 0 Theorem 5.2.11 and is a unique (t, r)-function. Exercise 5.4.9. Prove that if x E A and a E Min Tx then jxx l a ((a) , A) = 1. Describe the whole structure of the bisimulation jxx l a . 0 The notion of bisimulation allows us to introduce st-E-formulas which define, in [, the basic relations between coded sets in terms of A-codes. Index a still refers to "assembling" .
Definition 5.4.10. x II=: y is the st-E-formula " x, y E A Ajxy{A, A) = 1 " . X ae y is the st-E-formula " x, y E A A ({1) v {2)) " , where {1) x = "x and y = &.y for some internal x E y ; (2) Ty i:. {A} and there is b E Min Ty such that jxy {A, (b)) = 1 . (Note that these two cases are incompatible.) Finally, �t x is the st-E-formula 3s t y ( x = &.y ) (this implies x E A) ; B;in t x is the st-E-formula 3 int y ( x = &.y ) {this implies x E A) . 0
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The relations 8:: , ae, �t, IJ.nt have a pretty clear meaning in HST where the coded sets Ax do exist:
Theorem 5.4.11 ( HST ) . Suppose that x and y belong to A. Then {i) X 8:: Y iff Ax = Ay ; {ii ) X aE y iff Ax E Ay ; {iii) �t X iff Ax E S and IJ.nt X iff Ax E 0 . Proof. (i) As the map j : Tx x Ty -+ { 0, 1 } , defined so that j(t, r) = 1 whenever Fx{t) = Fy {r) and j(t, r) = 0 otherwise, is clearly a bisimulation, it coincides with jxy by Lemma 5.4. 10. This implies the required result. (ii) Suppose that Ax E Ay. H Ty = {A} then y = B.y, where y = y(A) = Ay is internal, and hence Ax is internal as well {because 0 is transitive), so that Tx = {A} because the code x is regular, therefore, x = "'x, where x = x{A) = Ax E y. H Ty f. {A} then clearly Ax = Fy {(b)) for some b E Min Ty, thus we have jxy{A, (b) ) = 1. This implies x ae y by {2) of Definition 5.4. 10. The converse can be proved the same way. (iii) Assume that Ax = y E S. Codes &.y belong to A (Lemma 5.4.6), and satisfy Aay = y (Lemma 5.3.6). It follows, by (i), that x 8:: B.y, hence, �t x. 0 The converse can be proved similarly. 5.4c The structure of basic relations
We continue to argue under the assumptions of 5.,4. 1. We are going to consider a = ( A ; ae , �t ; B:: ) as an invariant st-E structure. Then in particular we have to show that 8:: is an equivalence on
the domain A while ae and �t are �-invariant relations. In HST this task is pretty easy on the grounds of Theorem 5.4. 11. But the EEST case needs more work with codes and bisimulations.
Lemma 5.4.12. The bisimulations jxy { x, y E A ) satisfy the following: {i) jxx {t , t) = 1 for all t E Tx ; (ii) if r, t, p, r are flnite sequences such that r At E Tx and pAr E Ty {then, of course, T E Tx and p E Ty ) then j xl7' Y ip (t, r) = jxy(r At, p Ar) ; (iii) if t E Tx , r E Ty, u E Tz, then jxy{t, r) = jyz{r, u) = 1 => jxz{t, u) = 1. ,
Proof. (i) According to Definition 5.4.7, the set T of all t E Tx such that jxx{t, t) = 1 is inductive, that is, Max Tx � T and t E T provided any t A a E Tx belongs to T. Thus, T = Tx by the result of Exercise 5.3.2. (ii) The map j(t, r) = jxy(r At, p Ar) is clearly a bisimulation for the pair of codes xln YI P , and hence it coincides with jxl7' Y i p by the uniqueness. (iii) Consider this as a property of t E Tx {beginning with V r V u ) , say, P(t) . As T is a wf tree, it suffices to prove P(t) for any t E Max Tx and, if t ¢ M ax Tx , prove P(t) assuming P(t Aa) for any t Aa E Tx .
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201
First prove P(t) for t E Max Tx. In this assumption, jxy {t, r) = 1 implies r E Max Ty and x(t) = y(r), similarly, we have u E Max Tz and y(r) = z(u) , hence, x{t) = z(u) , now jxz {t, u) = 1 holds by Definition 5.4.7. Now suppose that t ¢ Max Tx. It follows from jxy{t, r) = jyz{r, u) = 1 that r ¢ Max Ty and u ¢ Max Tz. Consider any t Aa E Tx. By definition there exist r Ab E Ty and then u Ac E Tz such that jxy (t Aa, r Ab) = jyz(r Ab, u Ac) = 1. We conclude that jxz (t Aa, u Ac) = 1, by the assumption of P(t Aa). Thus, Y t A a E Tx 3 u A c E Tz Uxz (t A a, u A c) = 1) .
The same argument, in the opposite direction, shows that Y u A c E Tz 3 t A a E Tx Uxz(t A a, u A c) = 1) .
It follows, by definition, that jxz(t, u)
=
1, as required.
0
The proof of the lemma demonstrates the inconvenience of the absence of coded sets for arguments with A-codes in EEST : for instance, to prove (iii) in HST we can simply note that, say, jxy {t, r) = 1 is equivalent to Fx{t) = Fy {r). (This is similar to Theorem 5.4.11{i).) The equality Ax = {Ax la : a E Min Tx} (Exercise 5.3.4) is also meaningless in [ in any direct sense, yet we can attach an adequate meaning: ae-elements of any y E A are, modulo 11::::: , codes of the form Y l b , b E Min Ty , and only those codes:
Lemma 5.4.13. Suppose that x, y E A . Then (i) x 11::::: y iff either x = y = "'x for some internal x or Tx f. { A } f. Ty and {a) \;/ b E Min Ty 3 a E Min Tx (x l 11::::: Y l b ) , and {b) \;/ a E Min Tx 3 b E Min Ty {x l 11::::: Y l b ) . (ii) X ae y iff either X = "'x and y = 8.y for some internal sets X E y or Ty f. { A } and there is b E Min Ty such that x 11::::: Y l b . Proof. (i) Suppose that x 11::::: y, so that jxy {A, A ) = 1. If at least one of Tx , Ty is { A } then Tx = Ty = { A } and x{A) = y(A ) (Definition 5.4.7). Suppose that Tx f. { A } f. Ty, so that A ¢ Max Tx U Max Ty. Prove, for instance, (i) {a). Let b E Min Ty. It follows from 3* {a) of Definition 5.4.7 (with t = r = A ) that there is a E Min Tx with jxy {(a) , (b) ) = 1. Yet jxlaYib {A, A) = jxy {(a) , (b) ) by Lemma 5.4.12{ii), hence, x l 11::::: Ylb . As for the converse, if Tx = Ty = { A } and x{A) = y( A ) then jxy{A, A) = 1 by 1 * of Definition 5.4.7, hence, x 11::::: y. It remains to consider the "or" hypothesis of (i). We are going to prove jxy{A, A) = 1 applying 3 * of Definition 5.4.7. Let us check, say, 3* {a) of Definition 5.4.7 (t = r = A). Let b E Min Ty. Then, by {i)(a), there is a E Min Tx with jxla Yl b {A, A) = 1. As above, this implies jxy{ (a) , (b) ) = 1, as required for 3 * (a). (ii) By definition, it suffices to show that, for any b E Min Ty, x 11::::: Y l b is equivalent to jxy {A, (b) ) = 1. But the latter formula implies j x Yl b {A, A) = 1 by Lemma 5.4. 12{ii), as required. 0 a a
a
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5 Definable external sets and metamathematics of HST
Corollary 5.4.14. 11::::: is an equivalence on A while the relations 8E , �t, IJ.nt are 11::::: -invariant. Proof. 11::::: is an equivalence relation on A by Lemma 5.4. 12{iii). That the relation ae is 11::::: -invariant follows from Lemma 5.4. 13: for instance, if X ae y 11::::: y' then (in the nontrivial case Ty =f:. {A} ) we have x 11::::: Y l b for some b E Min Ty by 5.4.13{ii), on the other hand, Y l b 11::::: y' l b' for some b' E Min Ty' by 5.4. 13{i), and thus x B:: y' l 6, because B:: is an equivalence relation, thus X ae y' still by 5.4.13(ii). To see that the relations �t, 8j_nt are 11::::: -invariant apply Lemma 5.4.13(i). 0 5.4d The interpretation and the embedding
We continue to argue under the assumptions of 5.,4. 1. Definition 5.4.15. The results of Corollary 5.4.14 allow us to define an invariant st-E-structure
a =
( A ; ae , �t ; 11::::: ) .
0
Let � be the relativization of any st-E-formula � to e ( � a in the notation of § 1.5c). Thus � is the formula obtained from � as follows: (A) change all occurrences p = q, p E q, st p to resp. p 11::::: q, p 8E q, �t p; (B) relativize all quantifiers to A .
Lemma 5.4.16 (HST). The map x H Ax is a reduction of a t o the struc ture ( IL[O) ; E, st), where IL[O) = {Ax : x E A } . 5 Therefore we have {*) �{Ax p ··· , Axn ) L( O ) 8� {Xt , ... , xn ) { 8�{Xt , ... , Xn)) IE for any st-E-formula �( X1 , ... , Xn) and any Xt , ... , Xn E A .
Proof. The reduction claim follows from Theorem 5.4.11. The consequence is a particular case of Proposition 1.5.10. The rightmost equivalence follows from the fact that the domain A of the structure a is a subclass of [. 0 Theorem 5.4.17. (i) The structure a = ( A ; ae , �t ; 11::::: ) is an internal core interpretation of HST in EEST . {ii) (EEST ) The map x H &a; restricted to 0 is an internal core embedding of ( 0 ; E, st) (the internal core of the EEST set universe) into a, moreover, a st-E-isomorphism of ( 0 ; E, st) onto ( O (a} ; ae , �t) . {iii) (EEST) We have �(Xt , ... , Xn ) int 8(�{ Bxb ... , Ba;n) int ) for any st E-formula � and any (internal) sets XI J , Xn . The proof of the theorem will continue until the end of § 5.4e. First of all let us study properties of the map x H "'x : [ � A. We are going to prove slightly more than asserted by (ii) of the theorem, namely that the map st-E-isomorphically embeds [ onto a meaningful part of a. •••
5 The class li... [ D) will be considered in detail in Section 5.5.
5.4 From elementary external to all external sets
203
Definition 5.4. 18. A set x is sub-internal if it consists of internal elements. Accordingly subint x is the st-E-formula Vy (y E x => int y ) . 1P = { x : sub int x} is the class of all sub-internal sets. We define IP ( a}
= {x E A : {sub int x) a } = {x E A : V y E A (y aE x => y E o ( a} ) } ,
the sub-internal core of a .
0
Obviously [ � IP in HST and [ = IP = all sets in EEST .
Lemma 5.4.19. The map x � &a; : [ -+ A satisfies the following: (a) For any x E A, z E [ we have: {1) if x B: az and z is internal then simply x = az ; {2) if X ae az then there is X E Z (necessarily internal) with X = &a; . {b) For all X, y E [ : x = y iff ax B: &.y ; s t x iff as t &a; ; X E y iff &a; aE &.y ; int X iff IJ.nt ax . (c) We have: s(x) be a st-E-formula with codes in A as parameters. We have to find a code Y E A which ae-contains any code X ae X with aq)(x) , and (modulo 11:::: ) nothing more. If X = ax, X E 0, then let Z be the collection of all codes "'x, x E X satisfying aq)("'x) . If Tx ¥:. {A} then let Z be the collection of all codes of the form X I a, a E Min Tx, still satisfying ag>(XIa). Applying Lemma 5.4.20 to Z {which is a set-like collection by Theorem 5.2. 16} we obtain a code Y E A as required. Collection. Let X be a code in A and 4>{x, y) be a st-E-formula with codes in A as parameters. We have to find a code Y E A such that 3 y &g> (x, y) => 3 y ( y aE Y A &g> (x, y)) for any A-code X with X ae X. According to Lemma 5.4.20, it suffices to find a set-like collection Z � A such that (assuming that Tx ¥:. {A} )
(Y a E Min Tx) (3 y &q> (XIa, Y) => 3 y E Z &g> (XIa, Y)) . To get such a Z, just apply Theorem 5.2.16.
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5 Definable external sets and metamathematics of HST
Part 2: axioms for standard and internal sets
It follows from Lemma 5.4.19 that the st-E-structure of the internal core o ( a} of a is identical to the st-E-structure of the internal core 0 of [. Therefore ZFC st and Transfer of HST for a are immediate corollaries of the corre sponding axioms of EEST . Transitivity of 0 : follows from Lemma 5.4.19{c). Standardization. Let X be a code in A. Assume that Tx i:. { A } . Let A = Min T. Then D = {x E S : 3 a E A {XIa 11::::: ax) } is a set in [ {by Theorem 5.2. 14). Moreover there is a standard set S with D = S n S. If Tx = { A } , that is X = ax for some X E 0, then we let S be a standard set with X n S = S n S. In both cases, we have �t as, and moreover, the equivalence lla; E as "'x E X holds for any standard X . Regularity over 0 . Let X be a code in A, nonempty in a in the sense that there is at least one code in A which aE-belongs to X. We have to find another code X with X ae X such that any y E A which is an ae-element of both x and X satisfies IJ.nt y . We leave it as an exercise to show using Lemma 5.4.19 that if X = ax, X internal then a code x = "'x, where x is any element of X, is as required. Now consider the case when Tx i:. { A } . Then the set T = { t E Tx : 3 a E Min Tx
(XI t � Xla) }
is nonempty as well, for instance, Min Tx � T. As Tx is well-founded, there exists t E T such that none among the extensions t A b E Tx belongs to T. Let a E Min Tx witness that t E T. Then x = Xla (Example 5.3.4) is a code in A and X ae X by Lemma 5.4.13. 1f now (a) E Max Tx then X = ax, where x = X((a) ), so that x E o . Thus in this case X contains a IJ.nt-internal aE-element x. It remains to apply Transitivity of 0 in a . Suppose that (a) ¢ Max Tx. We claim that X n X = 0 in a . Let, on the contrary, a code y E A satisfy y ae X and y ae X. By Lemma 5.4.13 there is a' E Min Tx such that y 11::::: X I a' , and there is b E Min Tx such that y 11::::: x l b , which implies y 11::::: X l(a, b) · We conclude that X la' 11::::: X l(a , b) · t Ab' E Tx and Since Xl t 11::::: Xla and (a) ¢ Max Tx, there exists b' such that · X l t "b' 11::::: X l(a, b) · Then X l t " b' 11::::: Xla' , therefore t Ab' E T, contradiction. Part 3: axioms for sets of standard size
Note that Saturation (as defined in § I. H) is obviously relativized to the class 1P = { x : x � 0} (of sets which contain only internal elements). However it follows from Lemma 5.4.19 that the map sending every x E IP( a} to the unique x E [ with x 11::::: "'x is a reduction of the invariant structure ( IP < a> ; ae , �t ; 11::::: ) to ( [ ; E, st) {in the sense of Definition 1.5.9), and hence both structures have the same true st-E-statements. But Saturation holds in [ by Theorem 5.2.18.
5.4 From elementary external to all external sets
207
The following lemma demonstrates that the other two axioms of this group, Standard Size Choice and Dependent Choice, also are essentially rela tivized to the same class, although this is not immediately clear.
Lemma 5.4.21. For any code x E A there is a set D and a code f E A such that the following is true in a : "f is a function mapping D onto x ", where D = an E A . This lemma, together with the already verified axioms, shows that both
Standard Size Choice and Dependent Choice follow from the instances where
the domain of choices consists of internal sets. Thus the same argument as for Saturation above derives Standard Size Choice and Dependent Choice in a from the relevant results in [ {Theorem 5.2.14 and Exercise 5.2.19).
Proof (Lemma) . We can assume that Tx ¥:. {A} {the case x = ax for some x E 0 is rather elementary). Informally, as x is assumed to aE-contain x l a, a E D = Min Tx, as elements, we can map D onto x sending every a E D to x l a· To be more accurate, let, for any codes u, v in A, [u, v] denote a code p E A such that "p is a set containing u, v and nothing more" holds in the structure a = ( A ; aE , �t ; �). We put ( u, v ) = [[u, u] , [u, v]] , a code, in A, for the ordered pair ( u, v) = { { u } , { u, v} } . Let finally f be an A-code defined to aE-contain codes (aa, x l a ) , where a E D, and only them (as in the proofs of Separation and Collection). 0 0 ( Theorems 5.4.1 7 and 5. 1.4)
5.4f Superposition of interpretations To accomplish the proof of Corollary 5.1.5, we now prove its claims 4° , 5° by a rather straightforward superposition of the interpretations involved in the proofs of items 1 o , 2° , 3° of Theorem 5.1.4. Part 4 ° of C orollary
5.1.5
Recall that a = ( A ; aE , 8st ; �) is an invariant internal core interpretation of HST in EEST defined in § 5.4d {Theorem 5.4.17). Thus each of A, aE , �t, II:= is st-E-definable in the EEST universe, EEST proves that a is an invariant structure and proves 4> a for any axiom 4> of HST. Finally there is a map x H- "'x, provably in EEST a st-E-isomorphism of 0, the internal universe of the EEST set universe, onto the internal core o of a. Recall that e = ( E ; 8E , -st ; 8= ) is an invariant internal core interpreta tion of EEST in BST, § 5.2a. Each of E, 8E , -st, 8= is st-E-definable in the EEST universe, BST proves that e is an invariant structure, and proves 4> e for any axiom � of HST, and there is a st-E-isomorphism x H- 8X of the {internal) universe 0 of BST onto the internal core o<e> of e. {Theorem 5.2.6.) Arguing in BST , consider the superposition u of a and e . Thus u = (U ; ue , 'St ; U:: ) , where U = {p E E : e(p E A) } and for u, v E U :
208
5 Definable external sets and metamathematics of HST u � v iff e (u � v) ,
u ue v iff e(u 8E v) ,
"St v iff eC'st v ) .
Thus U, the domain of u, consists of those elements u E E which belong to the domain A of the structure a defined within e . The relations ue , 'St, � have a similar meaning, and hence in general u is a defined in e .
Proposition 5.4.22. u is an interpretation of HST in BST . Proof. This is based on the following claim: for any st-E-formula � with parameters in U, �u is equivalent to e(ag>) in BST. This can be proved by induction on the syntactical structure of �- For � an elementary formula this follows immediately from the definition of '1:= , ue , 'St. To carry out the nontrivial step for 3 let � be 3 x rp(x). Then �u is 3 x E U rpu (x). This can be converted, by the definition of U and the inductive hypothesis, to 3x E
E { e(x E A) A e(�(x))) , that is, to e(3 x (x E A A 8rp(x))) .
However the subformula in brackets in the right-hand formula is 8(3 x rp(x)) . Now let � be any axiom of HST; we have to prove �u in BST. By the above, it suffices to prove e(ag>). Since e is an interpretation of EEST in BST it remains to show that EEST proves aq>_ Yet this holds because a is an interpretation of HST in EEST . 0 Still arguing in BST, put � = e{Bx) for any x. In reality this means that is a function defined on the singleton { 0 } x { 0 } by � { 0 , 0) = { (A, x) }, where A, the empty sequence, is equal to 0 . (Recall that Bx = { (A, x) } for any internal x by Definition 5.3.5.) Let !li(x, y) be the formula y = { (A, x) }. Thus if x, y are internal then !li(x, y) expresses the equality y = Bx according to Definition 5.3.5. �
Lemma 5.4.23 ( BST ) . Let x be any (internal) set, p = ex, u = ux. Then we have e{int p A int u A !P{p, u )) . Less formally, u is equal to 8.p in e . Proof. Note that p E o<e> (see the second line in the proof of Theorem 5.2.6), in other words e{int p). By the same reasons u = � = e("'x) satisfies i e{int u ) . We observe that the formula !P{p, u ) is equivalent to !li (p , u ) nt in EEST provided p, u are internal. Thus it remains to show eyi(p , u) int . But this is equivalent to !li(x, Bx) by Theoorem 5.2.6(iii) since p = ex and o u = e( "'x). Finally !li(x, "'x) holds by definition. Note a remarkable inversion: � defined in the BST universe as e{Bx) turns out to be rather 8( ex) in e by the lemma. (In one and the same universe e("'x) = { ((0, 0) , { (A, x) } ) } and a( ex) = { (A, { ((0, 0), x) } ) } are ob viously different.) But this enables us to prove:
Proposition 5.4.24 { BST ). The map x � � is an internal core embed ding of the set universe ( 0 ; E, st) into the structure u = ( U ; ue , 'St ; � ), and moreover a st-E-isomorphism of ( 0 ; E, st) onto ( O ( u} ; ue , 'St) .
5.4 From elementary external to all external sets
209
Proof. The map x � ex is a st-E-isomorphism of ( 0 ; E, st) onto the in ternal core ( o ( e} ; eE , 'St) of e (Theorem 5.2.6). On the other hand e is an interpretation of EEST (still by Theorem 5.2.6), and hence by Theorem 5.4. 17 the map y � &.y defined in e is a st-E-isomorphism of ( O(e} ; eE , 'St) onto the internal core of the structure a defined in e. However the map y � a.y defined in e is just the map ex � U:z; by Lemma 5.4.23, while the structure a defined in e is by definition just the structure u = ( U ; ue , 'St ; U:: ) . It follows that the superposition x � ex � ux of the two maps is a st-E-isomorphism 0 of ( 0 ; E , st) onto the internal core ( O ( u} ; ue , 'St) of u as required. Part 5 ° of Corollary 5.1.5 To show that HST is standard core interpretable in ZFC we take the super position of the internal core interpretation u of HST in BST defined just above and the standard core interpretation *v of BST in ZFC defined in § 4.3c. The only notable extra issue is related to the fact that we require any standard core interpretation to be a structure with true equality in § 4.1b while both u and hence the superposition are invariant structures. But this discrepancy is immediately fixed by Theorem 1 .5.11. 0 (Corollary 5. 1.5)
5.4g The problem of external sets revisited Here we come back to the problem of external sets briefly considered in § 3.2f. The results of our study of metamathematics of HST (Theorem 5.1 .4, Corollary 5.1 .5) enable us to give a satisfactory solution to the problem of external sets in BST. Recall that the problem appears because many useful objects of study, for instance, st-E-definable parts (subclasses) of sets turn out to be not sets in BST, see § 3.2f. The solution given by Theorem 5.2.6 ( = 2° of Theorem 5.1.4) incorpo rates only those external "non-sets" which themselves consist of internal sets: recall that they were called "external sets ", § 3.2f. The theorem asserts that EEST is internal core interpretable in BST. Less formally, this means that BST is strong enough to build up a kind of external "envelope" or "hull" over its internal set universe 0, which turns out to be a much more complete universe of the elementary external set theory EEST ! In other words, BST contains full information regarding a large universe of external "non-sets" , including the opportunity to quantify over them. This advantage of BST (by the way, so far unknown for Nelson 's internal set theory 1ST ) is based on the parametrization theorem (Theorem 3.2.16). The existence of such an "enve lope" explains why somewhat na'ive "internal" considerations of external sets by 1ST practitioners are in fact consistent: those sets, mostly non-existing as internal sets in theories like 1ST or BST, are elements of a correctly defined envelope of "external sets" over the universe of all bounded sets in 1ST or the full universe of BST .
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5 Definable external sets and metamathematics of HST
The treatment of the external extension e in 0, the internal universe of BST, is in principle analogous to the treatment of complex numbers as pairs of real numbers. In other words, assuming that 0 is extended to e , the "universe" of all elementary external sets, the BST mathematician does not face any problem with uncertainty or illegality. Similar to the case of complex numbers, there is no need to translate everything back into the ground universe all the time, however the interpretation eq> (defined in § 5.2a) can be employed. Claim 4 ° of Corollary 5.1.5 provides us with a much more comprehensive solution. Not only external subsets of the internal universe of BST but all reasonable external sets of any kind (in particular those which contain other external sets as elements) can be consistently adjoined to the internal BST universe 0 in the form of an "envelope" or "hull" which satisfies axioms of HST ! In other words, a mathematician working in BST can legitimately assume that the universe 0 of BST is the internal universe of an external universe which satisfies the axioms of HST. This fact practically equalizes the bounded set theory BST with such an advanced theory as HST in the capability of treatment of external sets. Elements of the HST "envelope" can be visualized in the ground BST universe 0 by means of A-codes which are "external sets" from the 0-point of view. To explain what we have in mind in detail consider a couple of examples.
Example 5.4.25 (a monad) . Let IR denote the set of real numbers in 0, the universe of BST. A monad of a standard x E IR is the "external set" J.lx = {y E *IR x � y} (not a set in 0 ), where x � y means yst e > 0 {lx - Y l < e). We put Tx = {A} U { (y) : y � x} and Fx((y)) = y for any y � x. Then x is an "external set" {exercise: prove that it is non-internal). Moreover, x is a code in A and Ax = J.lx· 0 Example 5.4.26 (the set of all monads). Every monad is a bounded de finable class, so that this is still in the framework of "external sets" . However the collection of all monads is not a bounded definable class (of internal sets), therefore this is the point where the A-coding construction seriously enters the reasoning. We put Fx( (x, y)) = y for any x E IR n S and y � x, and :
Tx = {A} u { (x) : X E IR n S}
u
{ (x, y) : X E IR n s A y � X } ,
so that still x E A and Ax is the collection of all monads of standard reals. Saying it differently, x is the set of all monads of standard 0--reals in a. 0 One can develop in this manner in BST most of typical external construc tions of nonstandard mathematics. This is restricted only by properties of the theory HST itself, of course. Of those restrictions, the most notable is the fact that HST contradicts the Power Set axiom (see § 1.3b ). It will be shown in Chapter 6 how to define external universes which do satisfy Power Set, at the cost of the full Saturation axiom (which is replaced by Saturation restricted to a fixed cardinal).
5.5 The class IL[D) : sets constructible from internal sets 5.5 The class
IL.[D]
:
211
sets constructible from internal sets
Here we are going to pursue the HST case (see 5.4.1) for the considerations in Section 5.4. In other words, the structure a = ( A ; ae , �t ; 8:: ) (defined in § 5.4d) will be considered in the HST universe. Note that a set Ax does exist for any code x E A in HST. The class ll.. [ D) = {Ax : x E A} is studied in this section: by some reasons given below we call sets in ll.. [ D) sets constructible from internal sets. The class ll.. [ D) will be shown to be a transitive interpretation of HST having several additional properties unavailable in HST. For instance we prove that different infinite internal cardinalities remain externally different in ll.. [ D) - a result are not provable in HST. Another statement true in ll.. [ D) : for any cardinal "' all K--complete partially ordered sets are K--distributive in ll.. [ D) (in the absence of the axiom of Choi ce ! ). Blanket agreement 5. 5. 1 .
We argue in HST in this section. Accordingly
S, 0, [ indicate the classes of all resp. standard, internal, and elementary
external (as in Definition 5.2. 7) sets. Recall §§ 5.4a-5.4d on the class A of all regular A-codes x E [, the relations ae , �t, B.: on A and the a = ( A ; 8E , �t ; B.: ) . 0
5. 5a Sets constructible from internal sets The idea of relative constructibility is well known: following the ZFC pat terns, we should define as IL[D), the class of all sets constructible from internal sets, something like u� EOrd ILdD), where the initial level ll..o [D) = 0 consists of all internal sets, the union is taken at all limit steps, and any IL�+ l [D) consists of all sets st-E-definable in ILdD) - in particular, ll..t [D) = [. But in this case such an Ord-long inductive definition can be avoided: the following definition yields the same result (see Exercise 5.5.6).
Definition 5.5.2 ( HST ). We define ll.. [ D) = {Ax : x E A}, the collection of all sets which admit regular A-codes x E [. Sets i n ll.. [ D) are called sets constructible from internal sets. 0 It follows from Lemma 5.4.16 that the structures ll..[ D) and a have essen tially the same properties !
Exercise 5.5.3. Prove that the domain A/8.:: of the quotient structure a/B.: consists of 8.:-classes [x) a = {y E A : x B.: y} of codes x E A, generally speaking, proper classes. Apply {1) of Exercise 1.5. 7 to reduce the equivalence classes to sets as in the proof of Theorem 1.5.11. Prove that after such a reduction af�i:: will be st-E-isomorphic to ( ll.. [ D) ; E, st) . 0
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5 Definable external sets and metamathematics of HST
Theorem 5.5.4 ( HST ) . The class l[D) {that is, to be more precise, the structure ( l[D) ; E, st) ) is an interpretation of HST. In addition, 1° . [ UWIF � l[D), in other words, all elementary external, internal, standard, and well-founded sets belong to l[D) . 2° . l[D) is a transitive subclass of IH . 3° . Every set X E l[D) satisfying X � 0 belongs to [ . 4° . If a set X � l[D) is definable in l[D) by a st-E-formula with parameters in l[D) then X E l[D) . 5° . If a set X � 0 is definable in l[D) by a st-E-formula with parameters in 0 then X is definable in 0 by a st-E-formula with the same parameters.
6° . Every set X � l[D) of standard size belongs to l[D) . 7° . WIF is still the class of all well-founded sets in the sense of l[D) . go . In l[D) , every set is a functional image of a standard set. The primary goal of this section is to prove the theorem. In addition, we prove in § 5.5d that l[D) satisfies a useful transfinite form of Dependent Choice, most likely not available on the base of the axioms of HST . Technical arrangements in the proof of Theorem 5.5.4 will consist of trans formations of codes in A, mainly on the base of the following lemma.
Lemma 5.5.5 (HST ) . If a set Z � A is st-E-definable in [ (parameters in [ allowed) then there is a code X E A such that Ax = {Ax : x E Z} . Proof. The argument is pretty analogous to the proof of Lemma 5.4.20; we 0 leave it as an exercise for the reader. 5.5b Proof of the theorem on 1-constructible sets We begin with claims 1° - go of Theorem 5.5.4. 1 o. This is a consequence of Lemma 5.4.6: indeed the codes "'x, c["'v) (Definitions 5.3.5, 5.3.9} satisfy A-x = x, Ac["'v] = v (Exercises 5.3.6, 5.3. 10} . 2°. Suppose that x E X = Ax E l[D), where x E A. H Tx = {A} then X = x(A) E 0 by definition. Thus X � 0, and it remains to apply 1°. If Tx i:. {A} then x = Ax l a for some a E Min Tx, where xla is a regular A-code (Exercise 5.3.14} . Moreover, x la is st-E-definable in [ by a formula contain ing only x and a as parameters, where x and a belong to [, in fact a is even internal. Thus xla E A because [ interprets EEST by Theorem 5.2.11. It follows that x E l[D) . 3°. Assume that y E A and X = Ay � D. By 1.1. 11(3) there is a standard set s such that X � s. Then X = { X E s : "x ae y } by Theorem 5.4. 11 {because A-x = x }, and hence X is definable in [ by a st-E-formula with only y, S E [ as parameters. It follows that X E [ by Theorem 5.2.11.
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4°. Any x E X has the form x = Ax where x E A. According to the HST Collection , there is a set Z' � A such that such a code x can be chosen in Z' for any x E X. As in the proof of 3° above, the set Z = { c E Z' : Ac E X} is st-E-definable in [ (using only some elements of A as parameters) , thus, Z E [ by Theorem 5.2.11. Now apply Lemma 5.5.5. 5°. If X = {x E D : 4>L.[ 01 (x, y) } , where y E D (a parameter) then by (*) in the proof of Theorem 5.2.11 and in Lemma 5.4. 16 we have 6°. By definition there exist: a set S � S and a map g : S onto X. Using Extension {Theorem 1.3. 12) we obtain an internal function f with S � dom f such that, for each standard s E S, the set f(s) is a code in E satisfying EJ(s} E A and g(s) = AE f( • > . Then the set X = { EJ(s} : s E S} is st-E definable in D_[D) with sets S, f as parameters. It remains to apply 4°. 7°. That WIF � D_[D) follows from 1°. Further, a well-founded set obviously remains such in D_[D). If X E D_[D) is not well-founded then, by Dependent Choice, there is an infinite €-decreasing chain X 3 x0 3 x 1 3 x2 3 ... . As D_[D) is transitive, this chain belongs to D_[D) by 6°, where it still witnesses that X is not a well-founded set in D_[D), as required. 8°. Consider a set X = Ax E D_[D); x E A. The set P = Min Tx � D belongs to [ because Tx E [. In particular, P E and � D_[D). Note that X = { Ax l a : a E P} , and the map a 1---t Ax l a is st-E-definable in D_[D), and hence it belongs to D_[D) by 4°. Thus X is an image of a set P � D in D_[D). It remains to cover P by a standard set, using (3) of Exercise 1.1.11. As for the HST axioms in D_[D), the result in principle follows from The orem 5.4.17 (see Lemma 5.4.16) . However an independent proof on the base of 1° - 8° is very simple. The axioms of § 1. 1c, Regularity over D, and Satur ation are inherited from IH because WIFU D � [ � D_[D). To prove Standard Size Choice or Dependent Choice in D_[D) , we first get a choice function in IH. The function is a standard size subset of D_[D), so it belongs to D_[D) by 6°. As for the axioms of § 1.1b, all of them except for Collection are easy consequences of 4 °, and we leave this an an exercise for the reader. Collection. Since we have Collection in IH, it suffices to check the following: for any set X � D_[D) there is a set X' E D_[D) such that X � X'. Using Collection in 1H and (3) of Exercise 1.1. 11, we obtain a standard P with
\;/ x E X 3 a E P (Ea E A A x = AEa ) . The set P' = {a E P : Ea E A} belongs to [. Applying 4 ° as above, we easily prove that X � X' = { AEa : a E P'} E D_[D). 0 (Theorem 5.5.4 )
D_[D) as defined by 5.5.2 is equal to u�EOrd o_� [0) defined as in the beginning of § 5.5, as well as to the least transitive class which contains all internal sets and satisfies HST . 0 Exercise 5.5.6 (Difficult !) . Prove that the class
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5 Definable external sets and metamathematics of HST
5.5c The axiom of 0-constructibility Following patterns known from ZFC, we introduce the axiom of 0-construct ibility: let "IH = ll.. [ O)" be the statement: all sets belong to IL[O) .
Corollary 5.5. 7. "IH = IL[O)" is consistent with HST.
0
It is known from numerous set theoretic studies that Godel 's axiom of constructibility "V = L " allows to prove many results which ZFC alone does not prove, in particular, it greatly simplifies the structure of cardinals etc. The axiom "IH = IL[O)" plays a similar role in HST. The applications are mainly based on assertion 3° of Theorem 5.5.4 which allows us to extend properties of elementary external sets to all sets X � D. The next theorem gives some examples (see also Theorem 5.5.12):
Theorem 5.5.8. The following statements are consequences of "IH = IL[D)'' , therefore they are consistent with HST : {i) every cut (initial segment) U � "'Ord is standard size cofinal or standard size coinitial;
(ii) the axiom of Choice, in the form of § 1.1h, fails;
(iii) if X, Y E D and f : X ont� Y be any, possibly non-internal, function then *card Y � n *card X for some n E rN, in particular, if X is *-infinite then *card Y � "'c ard X. (Recall that *card is the cardinality in D. ) ; {iv) there exist infinite *-finite non-equinumerous sets; (v) every set X is either "large" or of standard size. (Recall that a set is "large" if it contains a subset equinumerous to an infinite internal set.)
Proof. {i) The set U belongs to [ by 3° of Theorem 5.5.4, hence, to A2 8 by Theorem 5.2.10. It remains to apply Theorem 1.4.6{i) . (ii) It follows from Theorem 1.4.7 that there is a partition of *rN which does not admit a A28 transversal, and hence does not admit a transversal of any kind under the assumption of "IH = ll..[ O)" . (iii) As in (i), f belongs to A2 8 • Apply Theorem 1.4.9. {iv) Let h E "'rN ' rN. Take internal sets X, Y with *card Y = 2h and *card X = h and apply (iii). {v) We have X = Ax for a code x E A. Let T = Tx and A = Min Tx, so that X = { Ax l a : a E A} {Exercise 5.3.4). It follows that X is equinumerous to the quotient A/ E, where E is an equivalence relation on A defined so that a E b iff xla � xlb· Yet both A and E belong to [ {because so does x), hence to A28 , so that the result follows from Theorem 1.4.1 1. 0 Problem 5.5.9. It follows from {ii) of the theorem that the negation of Choice is compatible with HST. Does HST prove the negation of Choice ? Is the negation of Choice given by the proof of (ii) the strongest possible ? o
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Exercise 5.5.10. Prove that the theory HST + "IH = IL[D) " is reducible to EEST in the sense that for any st-E-sentence gj there is a st-E-sentence cp such that HST proves gj cp[ . Argue as in the proof of 6 ° of Theorem 5. 1.4 0 in the end of § 5.2b using Lemma 5.4. 16. 5.5d Transfinite constructions in ll.. [ D) It was announced in the preamble to this section that ll.. [ D) models an ad ditional Choice-like property. The property we shall prove is, perhaps, not everything one can obtain in ll..[ D); one should try to prove for instance the existence of a maximal chain in each p. o. set. Nevertheless the one we prove will be of extreme importance in the development of forcing over ll.. [ D) below. Let us recall some notation related to ordered sets.
Definition 5.5.11. Let "' be a cardinal. A transitive relation is any structure P = ( P ; 2\ there are sets Q � P and G � 'Y with card G = 'Y such that P� = Q for all � E G, therefore, G � f(q) for any q E Q, so that *G � j(q) for any q E *Q by *-Transfer. Coming back to the parameter p, note that ·� E X = j (p) for any < � 'Y, hence p E *( P�) for any � < 'Y, and finally p E *Q. This implies *G � X = j(p) by the above, in particular, *G � J. It follows that *'Y � J as well because G � 'Y are sets of the same cardinality (see Exercise 6.1 .5(2)), which is a contradiction. 0 6.1c Simple relative standardness We consider here a rather special type of internal subuniverses: classes of the form S[w] = S({w}), where w is an internal set. By definition S[w] consists of all sets of the form j(w), where f E WIF is a function and w E dom j. 2
Note that this i s a generalization of (v) o f Theorem 6.1.3.
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Let w-s t x mean that x E S[ w ]. Sets in S[ w ] are called standard relative to w, or w-standard. Internal subuniverses of this type are interesting, in
particular, because some of them can be interpretations of BST� . The following is a list of basic facts related to classes of the form S[ w ] .
Exercise 6.1.8. Suppose that w E D. Prove the following: (1) S[ w ] coincides with the class S[ w ] defined earlier in 0 as a BST universe (Definition 3 .1 . 1 3); (2) S[w] is a thin class (apply 6.1.6{2)); (3) if X � S[w] then there exist a standard set Z and a map g E S[w) defined on Z such that X = {g(z) : z E Z n S} ; (4) the set N[w) = *N n S[w) is neither cofinal in *N nor coinitial in *N ' N ; {5) if w ¢ S then the set N [w ] is uncountable, moreover, if x E N[w] ' N then N [w ] n [0, x) is still uncountable. Hints. (3) By Standard Size Choice and Standardization, there is a standard set F such that any f E F is a function with w E dom f, and we have X = {f(w) : f E F n S}. Put Z = F and g(/) = f( w ) for f E F . (4) Apply Exercise 6 . 1.6(2) and (3) of Exercice 1.3.8. (5) Prove that N [ w ] is uncountable. Otherwise there is a function f : N2 -+ N such that N[w] = {j(n, w) : n E N } . Put g(n) = maxn' �n f(n', n). Then, for any n, we have g(k) > f(n, k) for all k ,2: n, therefore *g(w ) > *f(n, w) for any n E N by Transfer. However *g(w) E N[w ], contradiction. 0 Classes S[ w] admit a characterization in terms of ultra powers. Recall that if U is an ultrafilter over a set I then, for any set or class M, to obtain the ultrapower M 1 fU we define f � u g iff the set {i E I : f(i) = g(i)} belongs to U - for any f, g E M 1 ( M1 is the set or class of all functions f : I -+ M ), then put [/] = {g : f � u g} and define [/] *E [g) iff the set {i E I : f(i) E g(i)} belongs to U. Finally, M 1 fU = ( {[/] : f E M w } ; *E ) is the ultrapower. (See § 4.2b for detaifs.) Note that if I E WIF then any ultrafilter over I still belongs to WIF .
Proposition 6. 1.9. If w E 0 then there exists an ultrafilter U over a set I E WIF such that ( S[w] ; E) is isomorphic to WIF1 fU. Conversely, if U is an ultrafilter over a set I E WIF then there is w E *I such that ( S[w] ; E) is isomorphic to WIF1 /U . Proof. H w E 0 then there is a set I E WIF such that w E *I. Consider the set U = Uw = {X � I : w E *X} , the associated ultrafilter: one easily shows that U is an ultrafilter over I in WIF, and that ( S[w] ; E) and (WIF1 /U ; *E ) are isomorphic via the map sending any *f(w) E S[w] (where f : I -+ WIF, so that j : *I -+ S) to [/] E WIF1 fU. Conversely, given an ultrafilter U over a set I E WIF, we obtain, using Satu ration , a set w E 0 such that U = Uw . {Exercise: fill in the details of the proof.) 0
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6 Partially saturated universes and the Power Set problem
Exercise 6.1.10. Consider a class S(R), where R = {x� : � E �} � 0 is a set of standard size, � E WIF being a (well-founded) cardinal. By Extension there is an internal map x, dom x = ·�, with x� = x(*�), V � < �- There is an indexed family {D�}� < " E WIF of sets D� E WIF with x� E *D� , V �. Let U E WIF be the ultrafilter of all sets X � I = { (� , d) : � < � A d E D� } such that x E *X . Let § be the class of all f : I -+ WIF of finite support, i. e. there exists a finite set s � D such that f(y) = f(z) whenever y, z E I satisfy y(� , d) = z( � , d) for all � E s, d E D� . Prove the following generalization of 6.1.9: ( S(R) ; e) is isomorphic to the quotient power §fU . �See § 4.2b on quotient powers.) Why would the 0 modification with § = WIF not be satisfactory in this case ? 6.1d Gordon classes The notion of simple relative standardness admits an interesting modification which yields less sparse internal subuniverses:
Definition 6.1.11. Let w E D. Define rNM[w] = U rN[w] (the least initial segment of *rN containing rN [w] = S[ w] n *rN ) . Put SM[w] = S( { w} U rNM[w]) 3 (Gordon's classes). Say that a set x is w standard in the modified sense, in brief w-stM x, if it belongs to SM [w]. 0 Lemma 6.1.12. x E SM [w] iff there is a *-finite set y E S[w] containing x . Proof. Since x, w E 0 , there are well-founded sets X, W with x E *X , w E *W . The set K = rNM[w] satisfies (t) of 6.1.5(1). (For instance, if i, j E K then easily k = 2i 3i E K and i, j belong to S[k]. This argument
holds for any finite number of elements of K .) It follows from 6.1.5(1) that any x E SM [w] has the form x = j(w, n), where n E rNM [w] and f E WIF is a function satisfying (w, n) E dom *f. We can assume that simply dom f = W x rN. Further, n � m for some m E rN[w]. Take y = {j(w, n') : n' � m} , then x E y, y is *-finite, and y E S[w] by Theorem 6.1 .3(i). Conversely, assume that x E y E S[w], and y is *-finite. There is a function g E WIF with domg = W with y = *g(w). Further there are functions h, f E WIF, defined, resp., on W and W x rN, such that, for any w' E W, if g(w') is finite then h(w') = #g(w') and g(w') = {f(w', n) : n < h (w' ) } . 0 Then, by *-Transfer, x E y = *g(w) = {j (w , n) : n < *h (w ) } � SM[w] .
Corollary 6. 1. 13. (1) rNM[w] = SM [w] n *rN ; (2) if rN � rN[w] then rN � rN[w] � rNM[w] = SM [w] n *rN � *rN . Proof. If k E SM [w] n *rN then we have k E y E S[w] for a *-finite y � *rN. However n = sup y belongs to S[w], hence, to rN[w], and k � n. To see that the rightmost � in (2) is � apply (1) and Exercise 6.1.8(4) (non-cofinality). To see that the middle � is � use Exercise 6.1.8(4) (non-coinitiality). o 3 The subscript ,. will indicate "the modified sense, of simple relative standardness.
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Lemma 6.1.14. Let w E D. Any set X � SM [w] can be covered by a union of standard size many *-finite sets y E S[ w] . Proof. Note that by (3) of Exercise 1.1.11, there is a set C E WIF such that X � *C. The power set P = &(C) is well-founded, too. Let Y be the set of all *-finite sets y E S[w], y E *P, then Y is a set of standard size by the first part of the lemma. Finally X � U Y by Lemma 6.1.12: any *-finite set 0 y � *C is internal, hence, belongs to *P by *-Transfer. 6.1e Associated structures Given an internal subuniverse .JI, the following two related structures can be considered: type I : infiated standardness: ( D ; E, stJ ), where st.J1 x means x E .JI ; type II: deflated internal domain: ( .JI ; E , s t) . Either of them can be naturally viewed as a st-E-structure. (The atomic predicate st is interpreted as the unary relation st.J1 in ( D ; E, stJ), of course.) Since the internal universe D = ( D ; E, st) satisfies bounded set theory BST (Theorem 3.1.8), one can ask whether these structures still interpret any significant part of BST. s We observe that structures of both types satisfy ZFC t and Inner Trans fer by Theorem 6.1.3(i). Structures of type II obviously satisfy Inner Stan dardization, and some of them satisfy Basic Idealization restricted by a certain (well-founded) cardinal - their study follows in Section 6.2. On the contrary, structures of type I fail to satisfy Inner Standardization (except for two trivial cases), but those associated with Gordon classes do satisfy Basic Idealization . The proof of these two results follows in this subsection.
Theorem 6.1 .15. Suppose that .JI is an internal subuniverse that belongs to Def� . st (D), s � .JI � D. Then Inner Standardization fails in ( D ; E, St.J1 ) .
Proof. Case 1 : N � .JI n *N. It follows from Proposition 6.1. 7(ii) that the class X = { e E *Ord n .JI : e n .JI E D} is bounded in *Ord, therefore, X is a set. Accordingly, the least initial segment C � *Ord with X � C is a set. As .JI belongs to Def� . st (D) , so do X and C. It follows that C is a A28 set (in the HST universe of all sets) by Theorem 5.2.10. Suppose towards the contrary that Inner Standardization holds in the struc ture ( D ; E, stJ ). We claim that under this assumption the cut C is internal. Indeed otherwise the cut C is a gap by Theorem 1.4.6(i) , that is C has no maximal element while the complementary class *Ord ' C has no minimal element, and either C is standard size cofinal or else *Ord ' C is standard size coinitial. Let Y C *Ord be a set of standard size, cofinal in C in the first case, or coinitial in *Ord ' C in the second case. Choose any h E ( *N ' N) n
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6 Partially saturated universes and the Power Set problem
J. There is an internal set S � *Ord with #S :$ h, such that Y � S (Exercise 1.3.8). Thus S has stJ-standard number of elements in D. Then, applying Lemma 3. 1.18 in the structure (D ; E, StJ), we conclude that s n c is not cofinal in C and S ' C is not coinitial in *Ord ' C since C is a gap. This is a contradiction as Y � S . Thus C is internal, so that obviously C E *Ord. By definition X is a cofinal subset of C, satisfying X n � = J n � E D for any � E X. Theo rem 1.4.6(ii) implies that X itself is internal. Yet X = C n J - it follows that C E X, contradiction. 4 Case 2: Jn *N = N . 5 Suppose, towards the contrary, that Inner Standard ization holds in ( D ; E, stJ). Then J is a well-founded class: any non-empty set X � J contains an E-minimal element 6 • (Otherwise by a simple ar gument there is an €-descending sequence { Xn }ne N of elements of J. By Theorem 1.3. 12 (Extension) there is an internal function f, dom f = *N, with f(n) = Xn for all standard n. By Inner Standardization in ( D ; E, stJ), there is a function g E J such that g nJ = jnJ, in particular, g(n) = f(n) = Xn at least for all n E N. By the ZFC Regularity axiom in J, there is a number n E *N n J such that Xn+ l ¢ Xn· But *N n J = N, contradiction.) The well-foundedness allows us to define, by €-induction, a set F(x) E S for any x E J so that F(x) = 5{F(y) : y E x n J} (a standard set satisfying F(x)nS = {F(y) : y E xnJ} ). Then F is a 1 - 1 map (a simple argument by €-induction is left as an exercise), and x E y ¢::=:> F(x) E F(y) . In addition, F is definable in D, by a parameter-free st-E-formula. We claim that ran F = S. Indeed, as S is well-founded, it suffices to prove that s E S belongs to ran F assuming that s � ran F. Then the set X = { x E J : F(x) E s} (not necessarily internal) is st-E-definable in D (since so is F), therefore, by Inner Standardization in ( D ; E, stJ ), there is a set x E J with X = x n J. Obviously s = F(x) . Thus F : J ontS S is an €-isomorphism. It follows that G = F t S is an elementary embedding S � S, that is, for any E-formula 19(v1 , ... , vn) and any Xl J ... , Xn E S, we have '11 (x1 , ... , Xn) ¢::=:> '11 (F(x1 ), ... , F(xn )) in S. (This is because J is an internal subuniverse.) Moreover, G is not the identity because S � J. Finally, by (ii) of Exercise 3.2.4(3), G is €-definable in S. (Recall that F is st-E-definable in D. ) But this contradicts Kunen' s well known theorem (Theorem 6.1.21 below) . 0 4 Andreev [An 99] gave a somewhat simpler argument for the subcase rN � J n •rN � *rN. Then C = •rNnJ is a proper initial segment of •rN by Lemma 3.1.18{iv)
( applied in the structure ( 0 ; E, st.,sr ) ) . As in the main construction, C must be internal, thus it has, as any bounded subset of •rN, a maximal element, easily leading to contradiction. 5 This case is rather peculiar: it leads to measurable cardinals by Theorem 3.1.25. 6 The well-foundedness of J cannot be established here without the assumption of In ner Sta ndardization: a counterexample, not to be considered here, was com municated to the authors by Hrbacek.
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Let us study what happens with Idealization in structures of the form ( 0 ; E, w-st) and (0 ; E, w-stM) .
Theorem 6. 1. 16. Let w be an internal set. Then (i) the structure ( 0 ; E, w-stM) satisfies Basic Idealization ; {ii) the structure (0 ; E, w-st) satisfies � in Basic Idealization, but does not satisfy {x) to be !li 0"' (x), the relativization of !li to D " ), for any st-E-formula !P(x) with parameters in S(R) we have: 3 x E 0 " !P 0 �< (x) => 3 x E S{R) !li1"' (x) . 0 Proof (Theorem) . (i) Consider a set !£ � S(R) with card !£ S "- , n !£ i= 0. As in the proof of Theorem 6.2.3{i} , there exists a set u E WIF of cardinality S "' such that *u n n !£ i= 0. Let h be any map from "' onto u. Applying the K--completeness to the family of the 1-preimages of sets *u n X, X E !£, we easily obtain S(R) n n !£ i= 0 . (ii) We can suppose, that 4>(· ) contains only standard sets and some w0 E Rn , n E rN, as parameters, thus 4> will be written as 4>{x, wo ). We can further assume that 4>{x, ·) explicitly says that x is an *-ordinal and x < *"-, simply because S(R) � D" = S{*ll-) . In addition, it can be assumed by Theorem 3.2.3 that 4>(x, ·) is a E2t formula. Since S � S{R), the left most quantifier 3st in this formula can be eliminated, thus, let 4>(x, ·) be yst b rp{b, x, · ), where rp is an €-formula with standard parameters. After these simplifications, condition (ii){:f:) takes the form: if {1) there is an D-ordinal � < *"- such that yst b rp{b, �, w0 ) in 0 then (2) such an ordinal � exists in R. To prove this claim, we are going to restrict the variable b by a standard set of cardinality � *A in S, where A = 2" . As w0 E Rn � 0 " , there exists a set W E WIF such that card W S "' and wo E *W. Let, in D, Sb = {(�, w) E *"- x *W : • rp{b, �, w)}
for all internal b, so that 5b E D for all b E 0 since rp is an E-formula. Applying in 0 {which satisfies of ZFC ) the ZFC Collection and Choice, we get a set B of cardinality � *A in D such that V b 3 b' E B (56 = 56, ) . Such a set B can be chosen in S by Transfer. Then, by Transfer again, we have ystb 3 st b' E B (Sb = Sb' ) . This implies, in 0 ,
6.2 Partially saturated internal universes
235
We observe that, since B is a standard set satisfying card B � *A in S, there exists a surjection h mapping A = 2" (a well-founded cardinal) onto the set B n S. Now the last displayed formula takes the form: We define X11 = {� < � : rp{h{v) , �, w0 )} for every v < A. It follows from {1) that n il < .\ XII i:. 0. Then there exists � E R n nil < .\ XII {because R is A-complete). Now we have ystb rp{b, �, w0 ) by (* ) , i. e. (2), as required. 0 Let us now prove the existence of complete sets.
Theorem 6.2.11. Let "' � A < {} be (well-founded) infinite cardinals and {}-' = {}. Then for any set Ro � � of standard size with card Ro � {} there is a A-complete set R � *"- of standard size with Ro � R and card R � {} . Proof. Suppose that Q � R � �- We say that R completes Q iff Q � R and for every set {X11 : v < A} � S(Q) such that X11 � � for all v and X = n il 2" is a cardinal in WIF and F E IL[Dn), F : A � *IR is an injection. By definition there is p E Dn such that E, E A(Dn) and AEp = F, in particular, F is st-E-definable in ll.. [ D) with p as the only parameter. On the other hand, p, as any other member of Dn, has the form /{�}, where � < *"- is a *-ordinal and f is a standard function with � E dom f, by 6.2.2. In other words F is st-E-definable in ll.. [ D) with standard parameters and � as the only nonstandard parameter. Therefore so is every value F(a}, a < A (with the addition of one more standard parameter *a). It follows that F(a} belongs to the class S[�] = S{{�}), by Theorem 6.1.3{v} (for ..?' = S[�] ). Thus, by Standard Size Choice, for any a < A there is a map fo: : "' � IR (fo: E WIF, of course) such that F(a) = */o: (�). And, as F is injective, we have fo: i:. f"Y whenever a i:. 'Y < A. Therefore al � fo: is an injection of A into IR " , which contradicts the assumption that A > 2" . o
6.4 Partially saturated external universes
249
6.4f External universes over complete sets Let "' be an infinite (well-founded) cardinal. We demonstrated in § 6.2c that there exist sets R � *"- of standard size, called 2tt-complete, such that the corresponding internal subuniverses S(R) are elementary submodels of Ott in the st-E-language. Since the constructible internal core extensions ll.. [ Ott) and ll.. [ S(R)] are obtained on the base of resp. Ott and S(R) , it is a pretty natural question whether the external subuniverse ll.. [ S{R)) is a st-E-elementary sub model of ll.. [ Ott) in this case. The question should be answered in the negative in such a straight form, in particular, because the collection {of A-codes) A{..?' ) , which participates in the construction of ll.. [ J), is not defined by a st- E-formula relativized to ..?' but rather by a st-E-formula which refers to ..?' in a more complicated manner. Let alone the fact that internal subuniverses of the form ..?' = S(R) , where ..?' � Ott is a 2tt-complete set, are not necessarily self-definable classes, thus the results of § 6.4d are generally speaking not applicable. The goal of this subsection is to modify the definition of ll.. [ ..?') for internal subuniverses ..?' � Ott of this type so that the modified extensions of ..?' will be elementary substructures of ll.. [ Ott) in the st-E-language. Recall that [[Ott) = {E, : p E Ott} and ll.. [ Ott) = {Ax : x E A(Ott)} , where A(Ott) is, by definition, the class of all Ott-regular (in the sense of § 6.3b) A codes x E [(Ott) such that Tx � Ott, ran x � Ott. Now, for any internal subuniverse ..?' � Ott, let tt tt tt A(..?') = A{Ott) n [[..?') and ll.. [ ..?') = {Ax : x E A{..?')} . The class K.A {..?') is different from the abovedefined A{..?'), in particular, since now it is required that Tx � Ott and ran x � Ott rather than Tx � ..?' or ran x � ..?', yet we keep the condition of E-coding in ..?'.
Theorem 6.4.15. Let R � *"- be a 2tt-complete set. Then Kft.. [S(R)] is an
elementary substructure of ll.. [ Ott) in the st-E-Janguage, and hence an exter nal sub universe and an interpretation of HSTtt · In addition, Kft.. [ S(R)] is an internal core extension of S{R), that is Kft.. [ S(R)] n 0 = S{R) .
Proof. As ll.. [ Ott) � ll..[ O) is st-E-definable in ll..[ O) with only *"- E S as a pa rameter, it suffices to show that, for any st-E-formula 4>(x) with parameters in Kft..[ S(R)], if there is x E ll..[ Ott) satisfying 4>(x) in ll.. [ O) then such a set x exists in Kft..[ S(R)], or in a more formal manner (Compare with Remark 6.2.10 !) The key idea is to obtain the result required from Theorem 6.2.9{ii) (t), but in order to carry out this argument we use a reduction from ll.. [ O) to 0 (with [ as an intermediate structure) provided by the coding systems studied in Chapter 5.
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6 Partially saturated universes and the Power Set problem
Suppose that Yo E Kfl.. [S (R) ) is the only parameter in gj, thus, gj(x) is gj(x, Yo). By definition, we have y0 = Ay0 where y0 E "A(S(R)), and hence Yo = Eq0 for some q0 E S(R) n E. Now let tJ! {tJ , x, y) be the st-E-formula {}
E S is a *-ordinal A x, y E D... [ 019) A gj(x, y) ,
19
with the free variables {}, x, y. Let further 1/J(tJ, x, y) be the st-E-formula 8 ( 8(tJ!{tJ , x, y))). Then it follows from the equivalences ( * ) in the proof of Theorem 5.2.11 {the second equivalence) and in Lemma 5.4. 16 that
tJ!{AEn AEp , AE9 ) L( D ] { 8{tJ! {Et , E,, Eq )))[ 1/J{t,p, q) int ,
{2)
for all t, p, q E U = { u E E : Eu E A}. Note that q0 (see above) belongs to U and so does t0 = 8 ( 8 (*�)) {then ·� = AE to ) . Let us come back to our task. Suppose that x E D... [ O " ) witnesses the left hand side of {1), i. e. we have t/1 (*�, x, y0)L( D] . Then x = Ax where x E A{O " ), and hence x E [(0 " ), thus x = E, for some p E 0 " n E - and then p E U. In other words we have tP(AEto , AEp , A£ 90 )Ll01 , and hence 1/J(to ,p, Qo) int by {2). It follows from Theorem 6.2.9{ii){t) that the very same properties, that is p E 0 " n U, E, E A(O " ), and 1/J(t0, p, q0)1nt can be fulfilled by some p E S(R). But in this case x = E, E [[ S (R) ) , therefore x E "A(S{R)), and then x = Ax E Kfl.. [ S (R) ). Finally t/1 (*�, x, y0)L( D ] follows from {2), thus we have the right-hand side of {1} with this x, as required. That Kfl.. [ S(R)] n 0 = S(R) is left as an exercise for the reader. 0 Absoluteness
The major difference of the classes Kfl.. [ ..?') from those studied above {like D...[ J1) ) is that they are not transitive over the internal subclass J1 = Kfl.. [ J1) n 0 any more: indeed, this is because codes x E "A(J1) do not necessarily satisfy ran x � J1. This makes the absoluteness arguments outlined in § 6.3c not directly applicable for classes of this form. Yet under the conditions of Theorem 6.4.15 the absoluteness for Kfl.. [ S (R) ] holds to exactly the same degree as for the external subuniverse 0...[ 0 " ), which is an internal core extension of 0 " transitive over 0 " by Corollary 6.4.14. For instance, the formula " f is a function" is absolute for 0...[ 0 " ) by a general argument given in § 6.3c, but on the other hand Kfl.. [ S (R) ] is an elementary substructure of 0... [ 0 " ) w. r. t. this formula (and any other st-E-formula) by Theorem 6.4.15, and hence we can conclude that " f is a function" is absolute for Kfl.. [ S (R) ] as well. The reader can easily verify that many other similar simple formulas like being a pair, being a union, etc, are absolute for Kfl.. [ S (R) ] in virtue of the same argument. 1 9 If {) •"' where E WIF is a (well-founded ) cardinal then by definition D19 = =
K
D ��: and accordingly L[D19) = L[D ��: ). We prefer here to substitute the standard parameter •"' for the well-founded parameter "' E WIF.
6.4 Partially saturated external universes
251
6.4g Collapse onto a transitive class The transitivity of Kfl... [ ..?'] over ..?' can be restored by a procedure that re sembles Mostowski 's collapse known from works on ZFC. Recall that in ZFC any extensional set or class X admits a unique €-isomorphism f of X onto a (also unique) transitive set or class T. In the HST context, we prove
Theorem 6.4.16 ( HST ). Suppose that ..?' � 0 is an internal subuniverse and any internal set X � ..?' belongs to ..?' 2 0 Then (i) Any extensional class £ satisfying £ n 0 = ..?' admits a unique € isomorphism ¢ = t/>.J't' onto a class £Me transitive over ..?'. 21 (ii) £Me n 0 = ..?' and ¢ is the identity on ..?'. (iii) £Me � WIF[J] and £Me is a transitive internal core extension of ..?'. •
Proof. We define ¢(x) for all x E £ by induction on irk x, the rank over 0 (see § 1.5b}. If irk x = 0 then x E £ is internal, thus x E ..?'. Put ¢{x) = x in this case. Put ¢(x) = { ¢(y) : y E xn£} provided irk x 2: 1. In particular if irk x = 1 , SO that X � 0 but X ¢_ 0, then t/>{x) = x n J. We claim that ¢(x) is not internal for any x E £ ' ..?'. Indeed let x E ,ye , ..?' be a counterexample with the least possible irk x. Thus ¢(x) = {¢(y) : y E X n £} E 0, and hence any y E X n £ is internal. Thus ¢(x) = X n £ = X n ..?' E D. It follows that X n ..?' = X n £ E ..?' under the assumptions of the theorem, hence x = x n ..?' E ..?', a contradiction. We observe that ¢ is a bijection. Indeed if x E ..?' and y E £ ' ..?' then ¢(x) = x is internal while ¢(y) is not so by the above. Prove ¢(x) = ¢(y) => x = y for x, y E £ ' ..?' by induction on irk x, irk y. If ¢(x} = {¢(x') : x' E £ n x} = {¢(y') : y' E £ n y} = ¢(y) then £ n X = £ n y by the inductive assumption and hence X = y by the extensionality of £. Prove finally that ¢(x) E ¢(y) => x E y. Suppose that ¢(x) E ¢(y). If y E ..?' then ¢(y) = y is internal, thus so are both ¢(x) and x by the above, and hence x E ..?' and ¢(x) = x. If y ¢ ..?' then ¢(y) = {¢(y') : y' E .Ye n y} , thus ¢(x) = ¢(y' ) for some y' E £ n y , and x = y' E y by the above. We leave it as an exercise for the reader to prove the uniqueness of the collapse map t/>.J't' and the rest of the theorem. 0 Now, let "' still be a fixed infinite (well-founded) cardinal. Consider an internal subuniverse of the form ..?' = S(R), where R � "'" is a 2"-complete set of standard size. In this case the collapse map t/>.cn..[ S(R)] provided by Theorem 6.4. 16 for £ = Kfl... [S(R)] admits a pretty transparent 20 21
This condition is satisfied for instance if J is a thin class, see 6.3.4(2). Me means: Mostowski collapse. Note that &Me depends on J as well, not only on .Yt', but we suppress this dependence in the notation.
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6 Partially saturated universes and the Power Set problem
presentation in terms of A-codes involved. Indeed if x E "A(S(R)) then clearly X t S(R) = X r (domx n S(R)) is a code in A. But not only that. The following exercise contains a list of related results.
Exercise 6.4.17. Let x E "A(S(R)). Prove the following: ( 1 ) x and Tx belong to Kfl.. [ S(R)] . (2) Tx t scR> = Tx n S(R). (3) x t S(R) is a regular code and hence belongs to A . (4) H t E Tx n S(R), t Aa E Tx ( t Aa is an extension of t by a term a), and Fx (t Aa) E Kfl.. [ S(R)] then there is an extension t Ab E Tx n S(R) = Tx tscR > such that Fx (t Aa) = Fx (t Ab) . (5 ) Ax rs(R) = ¢"l[S(R) ] (Ax) · (6 ) The collapsed class (Kfl.. [S(R)]) Mc is equal to {Ax tscR> : x E "A(S(R))}. (7) (Kfl.. [ S(R)])Mc satisfies HST" together with Kfl.. [ S(R)] itself. Hints. ( 1 ) X belongs to [[S(R)] = {Ep : p E E n S(R)}, and hence is st-E definable with only p E S(R) as a parameter, and so is Tx, thus x and Tx belong to Kfl.. [ S(R)) by Theorem 6.4.15. (2) By a similar argument if t E Tx n S(R) then " t has an extension in Max Tx n S(R) " is absolute for Kfl.. [ S(R)) . (3) As S(R) is thin any internal set X � S(R) is finite. Then apply absoluteness as in ( 1 ) , (2). (4 ) : a similar absoluteness argument. 0 (5 ) follows from (4 ) . 6.4h Outline of applications: subuniverses satisfying
Power
Set
Theories HSTK and HST� contain less Saturation (and Standard Size Choice, for the first theory ) than HST does. However as soon as a particular applica tion is fixed, where all the cardinalities of sets involved are naturally bounded by a certain cardinal, the opportunitities offered by these partially saturated versions of HST are practically equal to those of HST ; in addition, HSTK and HST� contain the Power Set axiom ! To see how this can be used in the practice of nonstandard analysis, recall that in ZFC any particular mathematical structure 21 is a member of a certain transitive set W of the form V a E Ord. Usually we can take W = Vw+ w , the (w + w )-th level of the von Neumann hierarchy ( see § 1.5a) : indeed, all natural numbers belong to Vw = IHIF, so do all rationals, viewed as pairs of natural numbers, hence, all reals, defined by Dedekind, belong to Vw+ l , all sets of reals, including the set IR of all reals itself, to Vw+2 , all real functions appear at appropriate higher level, et cetera.) 0,
6.4 Partially saturated external universes
If, arguing in HST, we consider such a structure
253
21 in the class WIF of
all well-founded sets, then, accordingly, 21 belongs to a well-founded set of the form W = V0 , where a is a (well-founded) ordinal. Let us fix any (well-founded) cardinal "' � a as the amount of Saturation required to study 21 and its *-extension "m in suitable nonstandard manner. Something like "' = (card W) + will normally be sufficient. First option
The class IL[O " ] is a transitive internal core extension of 0 " satisfying HST" by Corollary 6.4. 14, that is, all of ZF (minus Regularity), together with such tools of the "nonstandard" instrumentarium as K--deep Saturation, hence also K--size Saturation (that is, Saturation for families of cardinality � K-) and the 2"-version of Standard Size Choice 22 and finally the Power Set axiom. Recall that the latter is incompatible with our basic theory HST itself. However any element of *V 0, including the set "m itself and all its el ements, belongs to 0" (because 21 E Vo � V" ), and hence to ll.. [ O" ). This allows us to carry out in ll.. [ O" ) any ordinary "nonstandard" argument related to 21 and "m which requires not more than the mentioned K--forms of Satur ation and, possibly, uses the Power Set axiom. If the results, obtained in the course of this study conducted in IL[O " ), are related only to 21 and "m and their elements, then they retain their meaning in the whole HST universe because both V0 and *V0 are transitive sets which belong to ll.. [O " ) together with all their elements. We call this approach the scheme ''WIF � 0" [ in ll.. [ O " ) ] ". Classes of the form "ll..[ S(R)) where R � *"- is a 2"-complete set of stan dard size (see § 6.4f) offer an additional opportunity. They are still internal core (non-transitive) extensions of 0 " satisfying HST" and elementary sub structures of ll.. [O " ) in the st-E-language. In addition they can be used to define consecutive extensions, see Remark 6.2.14. Second option
Classes satisfying HST� (with full Choice) can also be used. According to Corollary 6.4.4, there is a K--saturated thin internal subuniverse J � 0 " such that its internal core extension WIF[ J) satisfiess HST� including even full Choice (and Power Set), together with K--size Saturation. The class WIF[J] contains 21 and *2l, of course, but it is not true any more that *2l � WIF[ J), in fact, any set X � WIF[J) is a set of standard size. Yet one may employ the
22
That we have Saturation for families of cardinality K but Choice for domains of cardinality 2 ��: in IL[D ��: ] is remarkably in line with the practice of model theoretic nonstandard analysis, where it is customary to assume countable Sat uration (also called N1-Saturation) but sometimes to employ constructions which require continuum-many choices, see, e. g., the choice of TA in the proof of The orem 9.7.10 below.
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6 Partially saturated universes and the Power Set problem
absoluteness between WIF[ J1) and the entire universe, as in § 6.3c, to obtain, in the latter, an adequate meaning of facts established in WIF[ J1) . We can identify such a method as the scheme ''WIF --!..t J1 [ in WIF[ J1)) . "
An application
The following example, albeit rather elementary, shows how these schemes can be utilized. Consider, in HST, an infinite *-finite set H = [1, ... , h) � *N, where h E *N ' N. By Borel (H) they denote the least a-algebra of subsets of H containing all internal sets, which means, most naturally, the intersection of all a-algebras of subsets of H that include &int (H). But how to get at least one such a a-algebra? In ZFC there is no problem to take & (H), the power set. However &(H) is definitely not a set in HST for any infinite internal set H by Theorem 1.3.9, so that this argument does not work directly. Let us show how partially saturated subuniverses can be employed to solve this problem. Fix any infinite (well-founded) cardinal "'' for instance, "' = N0• It follows from Corollary 6.4. 14 that the subuniverse ll.. [ Dtt) satisfies HST tt, a rather rich partially saturated version of HST which includes the Power Set axiom. On the other hand *N � ll.. [ Dtt) still by 6.4. 14, in particular H E ll.. [ Dtt) and H � ll.. [ Dtt)· It follows that the power set PK = !Ji' (H) IL [ D,,:] , equal to & (H ) n ll.. [ Dtt) by the above, is really a set and belongs to ll.. [ Dtt)· Finally, we claim that PK is a a-algebra. Indeed recall that any set Q � ll.. [ Dtt) of cardinality � 2 tt in the HST universe belongs to ll.. [ Dtt) by Corollary 6.4.14. It follows that PK is even (2 tt )+ -additive, that is, closed un der unions and intersections of � 2 tt sets. Thus we have defined a a-algebra of subsets of H containing all internal subsets of H, and this is sufficient to consistently define Borel (H) . Historical and other notes to Chapter 6
Section 6.1. The notion of relative standardness (Definition 3. 1. 13) can be
traced down to [CherH 70) (in the context of the model theoretic nonstan dard analysis). Relative standardness, in the form of classes S[x) and SM [x], together with Theorem 6.1.16, is due to Gordon [Gor 89). Lemma 6.1.12 presents the original definition in [Gor 89). Thin classes: the definition is due to Andreev. Proposition 6.1.7 and Theorem 6.1.15 are due to Andreev and Hrba cek [AnH 04). The particular case N C *N n J1 � *N in Theorem 6.1.15 is due to Andreev [An 99) (also [Gor 89f for classes J1 of the form S[x] and SM [x] ). Hrbacek [Hr 01) explores more in this direction. The proof of Theorem 6.1.21 is based on ideas from [Suz 99). Theorem 68 in [Jech 78) gives a more general result, essentially saying that ZFCj, an extension of ZFC by a symbol j for an elementary embedding of the set universe in itself, with appropriate axioms, proves that j is the identity.
6.4 Partially saturated external universes
255
Section 6.2. Classes 0" (Definition 6.2.1) appeared in [Kan 91]. In the par ticular case "' = N0, sets that belong to countable standard sets were intro duced by Luxemburg [Lux 62] under the name of cr-quasistandard objects. A general definition was first given in a nonpublished version of Hrbacek [Hr 79], which the author of [Kan 91] was not aware of. The main parts of Theorem 6.2.3 appeared in our paper [KanR 95, Part 3]. The concept of A-complete sets and Theorem 6.2.9 appeared in [KanR 98] (partially in [KanR 95, Part 3] where the corresponding classes were denoted by 0 � ). Sections 6.3, 6.4. The content of these sections is mainly due to [KanR 95, part 3) ([KanR 97] contains an updated version), in particular, internal core extensions IL[O " ] and WIF[O�], introduced in [KanR 95] under the names, resp., IH" and IH� , and their main properties as in Theorems 6.4.3 and 6.4.13. Theorem 6.2.11, and applications similar to § 6.4f, appeared in [KanR 98]. See [Kun 80, 111.5] on the Mostowski collapse theorem in ZFC .
7 Forcing extensions of the nonstandard
universe .
Recall that the class IL[D) of sets constructible from internal sets was em ployed in Chapter 5 to obtain some consistency theorems. For instance The orem 5.5.8 implies that it is consistent with HST that 0-infinite internal sets of different 0-cardinalities are necessarily non-equinumerous. It would be in the spirit of mathematical foundations to ask whether the negation of this sentence, that is the existence of equinumerous 0-infinite internal sets of dif ferent 0-cardinalities, is also consistent. In ZFC, questions of this kind are often solved by forcing, 1 and it will be our goal in this Chapter to show how forcing works in HST. There are remarkable differences from the ZFC setting. First of all, the HST universe IH is not well-founded inside. This makes it difficult to define the forcing relation for atomic sentences by induction on the ranks of involved "names" , as in the ZFC case. We solve this problem us ing the well-foundedness of the universe IH over the internal universe D. This property allows us to treat IH as a sort of ZFC-like model with urelements; internal sets playing the role of urelements. Of course internal sets do not behave completely like urelements; in particular they participate in the com mon membership relation. But this gives us the key idea: generic extensions should not introduce new internal (and thereby new standard) sets. This leads us to another problem, connected with Standardization. Since new standard sets do not appear, a set of standard size cannot acquire new subsets in the extension. To obey this restriction, we apply a classical forcing argument: if the forcing notion is "standard size distributive" in the ground model then no new standard size subsets of IH appear in the extension. These ideas will be demonstrated on two examples. The first of them is a model of HST which "glues" 0-cardinalities of two given infinite internal sets having different cardinalities in the ground model. This example will be considered in Section 7.2 The other, a much more complicated example is a model of HST in Section 7.3, in which the isomorphism property (saying, in the context of HST, that any two elementarily equivalent structures of a language of standard size are isomorphic) holds. 1
We assume that the reader is acquainted with elements of forcing and has so me experience in it. Jech [Jech 78), Kunen [Kun 80], Shoenfield [Shoen 71) can be given as general references in this matter.
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7 Forcing extensions of the nonstandard universe
7. 1 Generic extensions of models of HST
This section discusses three principal elements of forcing in HST : the ground model, the forcing notion, and generic extensions. 7 .I a
Ground model
In Chapter 7 we argue in the ZFC universe V unless clearly stated otherwise. IH = ( IH ; E G-t , st G-t ) 2 is supposed to be a fixed model of HST, the ground model. We shall consider the well-founded, standard, and internal cores WIF = WIF( G-t ) = { x E IH : IH t= wf x} (all IH-well-founded sets), and s = s( G-t ) = {X E IH : StG-J X} (all IH-standard sets),
D = D (G-1) =
{X E IH : int G-t X}
(all iH-internal sets)
of the model IH, where int G-t x is the formula 3 y (st G-t y /\ x E G-t y). Unlike the case of models of ZFC, no model of HST can be an E-model in the ZFC universe simply because HST implies infinite €-decreasing chains of sets. Yet some regularity can be postulated. (a) All sets x E Do = D ' IHIF(G-t) (i. e. internal but not hereditarily finite in IH ) have one and the same von Neumann rank "' in the ZFC universe V, where "' is a cardinal > card IH . (b) If x E IH ' Do then the set x< G-t> = {y E IH : y E G-t x} is equal to x. Thus E G-t-elements of any set x E IH ' Do and €-elements of x in the universe is one and the same. In particular x � IH for any x E IH ' Do . 0 Blanket agreement 7. 1 . 1 .
Exercise 7.1.2. Prove the following, using 7.1.1: (1) x(G-1) = x for any x E WIF(G-1) (recall that WIF n D = IHIF in HST by 1.2.17). (2) IH is well-founded over D in the sense that the set Ord(G-t) of all iH-ordinals defined in IH as in § 1.2c coincides with the initial segment {� : � < ht G-t } of true ZFC ordinals, with one and the same order, where ht G-t E Ord, the height of IH, is the order type of ( Ord(G-1) ; E G-t ) . (3) WIF( G-1) is a transitive subset of Vhtt-t in the ZFC universe, and E G-t t WIF(G-1 ) coincides with E t WIF(G-t) . (4) If x E IH and y E IH ' D then the ordered pair p = (x, y) = { {x, y} {y } } belongs to IH ' D and IH f= p = (x, y) . (However we have p ¢ IH by 7.1.1(a) provided x, y E D and at least one of x, y belongs to Do .) (5) rN E IH ' D, and hence x = (x, rN) belongs to IH ' D and the equality x = (x, rN) is true in IH for any x E IH . 0 2 All HST-based arguments below will be restricted to this model, and hence it is rather convenient to denote it by IH, normally our symbol of the HST universe. The same applies to WIF, S, D just below in the text.
7.1 Generic extensions of models of HST
259
Requirement 7.1.1(b) can be interpreted as saying that any x E IH ' Do is a true set in the sense that it is in the universe what it seems to be in IH : the elements are the same. On the other hand sets in Do are just IH-sets: their true elements may have nothing to do with E1H-elements. It follows from (3) that natural numbers and hereditarily finite sets in IH are equal to those in the ZFC universe. In particular 0 E IH and 0 ( 1H} = 0 , so that 0 still is the IH-empty set. Moreover the set N = N( IH} E IH is equal to N ( IH} (that is, N in the sense of IH ). Saying it differently, 0 and N are absolute for IH. Assertion (4), an easy consequence of (1), says that the operation of ordered pair is absolute for IH as well, provided at least the second term of the pair considered is not IH-internal. There are other simple absoluteness results, for instance "being a subset of X x Y" provided X, Y E IH and Y � IH ' D, not mentioned in 7. 1.2. Requirement 7.1.1 (a) looks rather artificial; but we make use of it in the proof of Lemma 7.1 .10.
Exercise 7.1.3. Prove that any x E Do satisfies x � IH, even x n iH = 0 . 0 Exercise 7.1 .4. Prove that any model IH of HST well-founded in the sense of 7. 1.2(2) is isomorphic to a model satisfying 7.1.1 and then 7.1.2, 7.1.3. Hint. 7. 1.1 (a) can be assumed immediately. To achieve 7.1. 1(b) define f( x ) = x for x E Do and f( x ) = {f(y) : y E IH x} otherwise. This is sound because of 7.1.2(2) and since IH ' Do = (IH ' D ) U IHIF is well-founded in HST. 0 The /-image of IH is as required. Thus the real content of 7. 1.1 is the well-foundedness of IH over its internal core while the rest is just simple cosmetical rearrangements.
7.1b Regular extensions Forcing is a powerful method that allows to extend models of certain theories to models (of the same or a closely related theory) which have some desired additional properties. Let us formulate some basic requirements to be satisfied by such an extension in the case of models of HST .
Definition 7. 1.5. A st-E-structure IH ' = ( IH ' ; E IH' , stiH' ) is a regular exten sion of a model IH = ( IH ; EIH , stiH) satisfying 7.1.1 if (1) IH � IH ' , E IH is equal to the restriction E IH' t IH, and IH is an E1H'-transitive subset of IH', (2) the classes s < IH' ) , D (IH' } (standard and internal sets in IH ' ) coincide with resp. S = S(IH} , D = D(IH} , and finally (3) for any x E IH ' ' IH we have: a) x = x 0. Then we have two subcases: a) if the set aput [G) = {c[G) : 3 p E G ((p, c) E a) } (a putative G interpretation of a) is equal to x ( G-t) for some (unique by Extensio nality in IH ) x E IH then put a [G) = x ; b ) otherwise define a [G) = aput [G) (the true G-interpretation of a ) . Note that even in case 2a a [G) = aput [G) holds provided the unique set x E IH satisfying aput [G) = x ( G-t ) belongs to IH ' 0 because then x ( G-t) = x by 7.l .l(b). A sufficient condition for this subcase is aput [G) � D . We put IH[G) = {a[G) : a E Nms(P) }. Define the membership EG in IH[G) as follows: y EG x in either of the two following cases: 3 IN can be replaced by any fixed non-internal set in IH in the defintion of x; all -
we need is that
x �
x is an injection IH -+ IH ' D .
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7 Forcing extensions of the nonstandard universe
A) x, y belong to IH and y ED-i x, B) x ¢ 1H and y E x in the ZFC universe - thus EG-elements of any x E IH [ G] ' IH and €-elements of x is one and the same. Define the standardness stG in IH[G) so that stG x iff x E IH and x is standard in IH; thus stG coincides with stD-l . This completes the definition of the model IH[G] = (IH[G] ; EG, stG) · Suppose that a, b E Nms (P) and p E P. Define a preliminary forcing relation, only for atomic formulas of the form b E a, as follows:
p fore b E a iff
{ 3y
E x ( b = y) whenever a = x E Nmso ; 3 q 2: p ((q , b) E a) whenever nrk a > 0 .
The next lemma explains in more detail how the membership in the extension is organized in terms of fore . Lemma 7.1 .10. Assume that a E Nms(P ) while G � P is a is a filter in the sense that p E G ==> q E G whenever p, q E P and q ;::: p. Then for any y E IH[G] each of the following conditions (i), (ii) is equivalent to y EG a[G] :
(i) a) y E x(D-l ) = {y E IH : y ED-i x} , provided a = x E Nms0 , b) y E aput [G] , provided nrk a > 0 ;
(ii) 3 b E Nms (P) 3 p E G (y = b[G] 1\ p fore b E a) . Proof. Since (i) (ii) is an immediate corollary of the assumption that
G is a filter, we can concentrate on the equivalence y EG a[G) (i) . The only possible counterexample to this equivalence is a name a E Nms(P) with nrk a > 0 such that the set x = aput [ G] belongs to IH but x i:. x(D-1) , and hence x E Do by 7.1.1. It follows from our definitions that x = aput [G] is the result of an as sembling construction, of the type considered in Section 5.3, which begins with sets in IH and contains at least one step (since nrk a > 0) but has a total height nrk a < htD-l. If the initial sets of the construction all belong to IH ' Do then x = a( G) is a set of the von Neumann rank � htD-l + htD-l < K by 7.1.1(b}, thus x ¢ D0 by 7.1.1, contrary to the above. Thus at least one of the initial sets belongs to D0• But then the result x = a(G) is a set of the von Neumann rank > K in the universe by 7.1.1(a), and hence x ¢ IH by 0 7.1.1, still a contradiction. Corollary 7.1 .11 . Suppose that G � P. Then of IH satisfying 7.1 .2(2) and htD-l ( G ) = htD-l . If in addition P E IH then G E IH[G) . 4
IH[G) is a regular extension
4 Theorem 7.1.20 below shows that IH[G] satisfies HST for a wide category of sets G � P (generic sets) provide d P itself satisfies certain requirement s.
7.1 Generic extensions of models of HST
263
Proof. To prove IH � IH[G) note that x[G) = X for any X E IH and G i= 0. To verify the transitivity of IH in IH[G], assume that x E IH, y E IH[G], y EG x. Then by definition X = x[G) where X E Nmso , and hence y belongs to the set x(D-i} by Lemma 7.1 .10, that is y E IH and y E G-t x . If P E 1H then G = { (p, jJ) : p E P } is still a set in IH, and moreover G E Nms (P) by 7. 1 .9, while on the other hand G[G) = G for any 0 f. G � P ! (This argument obviously fails if P is a proper class in 1H .) Prove 7. 1 .5(3)b. Suppose that a E Nms(P) and a[G) E IH[G) ' IH, and hence a ¢ Nmso . Then a[G) f. z(G-t} for any z E IH because otherwise we would have a[ G) = z E IH by definition 2a. The rest of the corollary is left as a simple exercise for the reader. 0 We finish with two boundedness-type results. Define, for any a E Nms(P) , 6.a =
{ {y
E x}
whenever a = x E Nms 0 , ran a = {b : 3 q ((q, b) E a}} whenever a E Nms(P) ' Nms 0 • : y
Exercise 7.1 .12. (1) Prove that 6.a E IH, 6.a � Nms(P), and p fore b E a implies that b E 6.a and either a, b E Nms0 or nrk b < nrk a. (2 ) Prove that if a E Nms(P) , G � P, and x EG a[G] then x = b[G) for some b E Nms (P) satisfying b E G-t 6.a . 0 Lemma 7.1.13. For any name a E Nms(P) there is a set s(a) E S such that we have y E G-t s(a) whenever y E 0, G � P, and y EG a[ G) . Proof. Define s(a) in IH by induction on nrk a. If a = x E Nms0 then x n 0 can be covered by a standard set by the axiom of Boundedness in IH, and hence there is a standard *-Ordinal € Such that X n 0 � V� (the €-th level of the von Neumann hierarchy in D ) . Put s(a) = V� , where € is the least standard *-ordinal of this sort. By definition any y E 0 with y EG a[ G) = x satisfies y E s(a) in IH. If nrk a > 0 and s(b) E S is defined for all b E 6.a then put, in IH, s(a) = &int (V� ) (the power set in 0 ; s(a) is standard together with € and the set v� by Transfer) where € the least standard *-Ordinal with U b e 6a s(b) � v� . Suppose that y E 0 and y EG a[ G). Then, in IH, y = b[G) for some b E 6.a by 7. 1. 12(2), and hence s(b) � V� , then y n D � V� by the inductive assumption, y � V� as 0 is transitive in HST, and finally y E s(a) = &in t (V� ) . 0 7.1e Forcing relation
We continue to argue under the assumptions of 7. 1. 1, 7. 1. 7. Definition 7.1.14. A P-forcing relation is any relation p I � � whose argu ments are conditions p E P and closed st-E-formulas � with parameters in Nms (P), satisfying the following requirements F1 - F7:
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7 Forcing extensions of the nonstandard universe
Fl: For any x, y E IH : p I� y = x iff y = x, and p I� y E X iff y E X. F2: p I� a = b iff for each condition q � p and every name c E Nms (P} : 1} q fore c E a implies q I� c E b, and 2) q fore c E b implies q I� c E a . F3: p I� b E a iff for each condition q � p there exist r � q and a name c E Nms (P} such that r fore c E a and r I� b = c. F4: F5 : F6: F7:
p I� st a p I� ..., 4> p I� (4> A !P} p I� V x 4>(x)
iff iff iff iff
v q � p 3 r � q 38t S (r I� a = s) .
none of stronger forcing conditions q � p forces 4> . p I� 4> and p I� !li . p I� 4>( a ) for every name a E Nms (P} .
In the whole scheme Fl - F7 p,
q,
r
are forcing conditions in P.
0
Fl obviously implies both F2 and F3, and hence there is no need to stress that at least one of the names a, b does not belong to Nms0 in F2, F3. Items F4 - F7 handle st and non-atomic formulas. It is assumed that other logic connectives are combinations of A, V . •,
Theorem 7.1.15. Under the assumptions of 7. 1.1, 7. 1.7, there exists a
unique P-forcing notion, denoted by I�P henceforth. This forcing notion satisfies the following de/inability requirements, in which rp (x 1 , ... , xn} is an arbitrary st-E-formula:
(i) If P E IH (set-size forcing} then the relation p I�P rp (a 1 , ... , an} , with p, a1 , ... , an as arguments, is st-E-definable (with parameters in IH al lowed, including P as a parameter, of course) in IH, i. e. the set is is st-E-definable in IH (with parameters in IH allowed}. (ii) Moreover, the relation p I�P rp (a 1 , ... , an) with the arguments p, P , and a1 , ... , an, is also st-E-definable in IH (parameters allowed}, i. e. the set
{ (P , p, a1 , .. . , an ) : P E IH is a p. o. set satisfying 7.1 . 7 A p E P A a1 , .. . , a n E Nms (P} A p I�P rp (a 1 , . . . , a n) } is st-E-definable in
IH (parameters allowed}.
The definability of the forcing relation �P in the case when the forcing notion P is a proper class in the ground model is too complicated an issue to be considered here. In the only example of such a "class" forcing, studied below in Section 7.3, the definability will be obtained by reduction to set-size subforcings.
7.1 Generic extensions of models of HST
265
Proof. To prove the existence and uniqueness we have to show that F1,
F2, F3 form a legitimate scheme of well-founded induction. Let g_j be the collection of all formulas of the form b = a and b E a, where a, b E Nms (P) . For any cp E g_j let R'P indicate the collection of all formulas 1/J E qi to which the definition of p I� cp can directly refer according to F2 and F3 for different p E P. To be more precise, 1) if cp is iJ = x or iJ E x, where x , y E IH, then RIP = 0 ; 2) if cp is a = b where a, b E Nms (P) and at least one of a, b does not belong to Nms o then RIP consists of all formulas of the form c E a where c E �a and all formulas c E b where c E �b (see 7.1.12 on �a, � b ); 3) if cp is b E a where a, b E Nms (P) and at least one of a, b does not belong to Nms o then � consists of all formulas b = c where c E �a •
Define a partial order -< on qi as follows: cp -< 1/J iff there exists a finite sequence cp = cpo, cpl , ···, cpn = 1/J ( n � 1 ) such that 'Pk E �lo+l for all k < n.
Lemma 7.1 .16. -< is a well-founded partial order on
qi .
Proof. Let, on the contrary, cpo >- cp1 >- cp2 >- . . . be an infinite decreasing chain in g_j, so that 'Pn+l E RIPn , V n. Assume that cpo has the form a = b.
(Otherwise 'Pl has such a form.) Then 'P2n is an = bn for all n, and moreover, by definition, either an+l , bn+ l E �an or an+ l , bn+l E � bn for any n. Thus either some cp2 n is of the form x = iJ - and then the chain breaks because formulas of the form x = iJ and x E iJ are - •
•
is an internal set for any £00-formula rp{x1 , , X m , yt 1 , , ytn ) with parame ters in �[D). (Hint. Argue by induction on the complexity of rp . The corre sponding transformations of sets X'P are absolute for 0 by 1.2.5.) 0 ••.
•••
Suppose that p is an internal 1 - 1 map from an internal set D � A onto a set E � B {also internal). We expand p to all types t by induction: for all xt E Dt , whenever t = o(t1 , ... , tk ) . Then pt internally 1 - 1 maps Dt onto Et . If � is an £00-formula containing parameters in D00 then let p g, be the formula obtained by changing each parameter X E nt in g, to pt (x) E Et . 13 Clearly Dt 1 n Dt 2 f. 0 for different types h , t2 ; for instance 0 belongs to any D t , t f. 0. However it is supposed that appropriate provisions are taken to distinguish equal sets which appear in different types.
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7 Forcing extensions of the nonstandard universe
Definition 7.2. 10. Under the assumption (t) of Theorem 7.2. 7, let 1P .c {2l, �) be the set of all internal 1 - 1 maps p such that D = domp is an (internal) subset of A, E = ran p � B {also internal), and, for each closed £00-formula � with parameters in D U D00, we have: 2l[D) t= � iff �[E) t= p � . 0 We define p � q (p is stronger than q) iff q � p .
For instance the empty map 0 belongs to IP .c {2l, �) because 2l and � are elementarily equivalent. {Proper £00-variables can be eliminated in this case because the domains 0t are finite.) Lemma 7.2.11. Under the assumption 7.2. 7{t) , the p. o. set IP = IP .c (2l, �) is standard size closed and hence standard size distributive by " IH = ll.. [ D) " . Proof. Let A be a limit ordinal. Suppose that p0 , a < A, are conditions in IP, and Pf3 :$ Po whenever a < {3 < A. We claim that IP is II� s , i. e. a standard size intersection of internal sets. Indeed by definition IP = n"' P"' , where cp
runs over all parameter-free £00-formulas having no proper £-variable free, and, for any such formula cp(yt1 , , yt n ), P"' consists of all internal partial 1 - 1 maps p A -+ B satisfying, for all yt; E nt; , 1 :$ j :$ n , •••
:
where D = domp and E = ranp (internal subsets of resp. A , B ) . However there exist only standard size many parameter-free £00-formulas cp, and on the other hand all sets P"' are internal by 7.2.9 {and because all maps pt are internal together with p) . Therefore all sets Wo = {p E IP : p :$ Po } are II� s as well. Furthermore W0 i:. 0 and we have W13 � Po whenever a < {3 < A. Finally A (as every set in WIF, the well-founded universe) is a set of standard o size by Lemma 1.3.1, thus no< " Wo i:. 0 by Theorem 1.4.2{i). 7.2e Key lemma
The next lemma is of key importance since it shows that IP = IP .c {2l, �) be haves more or less like a collapse forcing, in particular, a condition p E IP cannot satisfy both dom p � A and ran p = B (or vice versa) . The difference from typical collapse forcing notions is that IP .c {2l, �) still can contain mini mal {that is, strongest and �-maximal) conditions p, but any such condition p satisfies dom p = A and ran p = B by the lemma, and hence immediately exhibits an isomorphism between 2l and � already in M. It follows that there is no need to explicitly require that dom p has *-Cardinality less than A and B, as in § 7.2a with respect to collapse forcing notions. Theorem 7.2 .12 (the key lemma) . Still under the assumption 7.2. 7(t) , suppose that p E IP = IP .c{2l, �), D = domp, E = ranp. If a E A ' D then there exists b E B ' E such that P+ = p U { (a, b) } E IP. Conversely, if b E B ' E then there exists a E A ' D such that P+ = p U { (a, b) } E IP .
7.2 Applications: collapse maps and isomorphisms
277
Proof. By symmetry, we concentrate on the first part. Let us fix a condition p E 1P. Let D = d om p, E = r an p. {For instance we may have p = D = E = 0 . ) Consider an arbitrary a E A ' D; we have to find a counterpart b E B ' E such that P+ = p U { (a, b) } still belongs to IP. Let "' = card £ {or "' = No provided .C is finite). Let us enumerate by rt'o (x, x ta ) (a < K-) all parameter-free £00-formulas containing only one free £-variable x and only one free £00-variable xta of a type t0 ¥:. 0 . We define Xo = {xta E Dta : 2t[D) F rp0 {a, xta ) } for all a < K-; thus Xo E Do( ta } . {Note that Xo is internal by 7.2.9.) Let !li0 {X, x) be the £00-formula V xta (X(xta ) rp0 {x, xta )), thus 2t[D) F !li0 {X0 , a) by defi nition. We have K--many formulas !li0 {X0 , x) realized in 2t[D] by one and the same element x = a E A. We put Y0 = po( ta } (X0); thus Y0 E Eo( ta } . Lemma 7.2.13. There exists b E B such that �[E)
F !li0 {Y0 , b) for all a .
Proof. It suffices to prove that any finite conjunction !li01 (Y01 , y) A · · A !Pam (Yom , y) can be realized in �[E) . By definition a witnesses that ·
2t[D]
t= 3 X {!P (X ol
ol ,
x) A . . . A !Po m (Xo m , x)) .
Therefore �[E) F 3 y {!li01 (Yap y) A . . A !Porn (Yom , y)) , since p E IP . ·
0
Let us fix an element b E B satisfying !li0 {Y0, b) in �[E) for all a < "-· We set P+ = p U {(a, b) } , D+ = D U {a} , E+ = E U { b } . We claim that P+ E IP, i. e. the equivalence 2t[D+] t= � iff �[E+ ] t= P+ � holds for each closed £00-formula � with parameters in D+ U D+ 00 • The next lemma gives a partial result. Lemma 7.2.14. Let rp(x) be an £00-formula possibly with sets in D00 parameters. Then rp(a) is true in 2t[D] iff {prp) {b) is true in �[E) .
as
Proof. Assume that rp contains only one parameter � E nt and t ¥:. 0 (otherwise use a tuple coding). Then rp{x) is rp0 {x, �) for some a < "' such that t = t0 • Since !li0 {X0 , a) is true in 2t[D), we have: X0 (�) iff 2![D] F rp0 {a, �) . Note that X0 (�) Y0 (1J) where 1J = pt (�) E E t because p E IP. On the other hand we have Y0 (1J) iff �[E) F rp0 (b, 1J), because !li0 (Y0, b) is true in � [E) by the choice of b. But rp0 (b, 1J) coincides with (prp) {b) . 0
However there is a more serious problem: we have to check that P+ trans forms true £00-formulas with parameters in D+ 00 into true £00-formulas with parameters in E+ 00 • The idea is to convert formulas with parameters in D+ oo (not necessarily equal to a) into formulas with parameters in D00 plus a as an extra parameter, and use Lemma 7.2.14. Fortunately the structure of types over an internal set C depends only on the internal cardinality of C but does not depend on the exact choice of C. This allows to "model" D+ oo in D00 identifying the a with 0 and any a e D with {a} . To realize this plan, let us define U = { 0 } U { {a} : a e D} ,
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7 Forcing extensions of the nonstandard universe
so that U � D\ where -r = o(O) (the type of subsets of D ) . Furthermore we have u E no(T) because u is internal. Accordingly, on the other side, we put V = {0} U { {b} : b E E}; then V � E\ V E Eo(T) , and V = po ( T) (U) . For each type t, we define a type b'( t ) by b'{O) = -r and b' {t ) = o{b'{ti ) , ... , b'(tn)) provided t = o (t 1 , .. . , tn) · Put b'{a) = 0, and b' {a ) = {a} for all a E D , so that b' is an internal 1 - 1 map from D+ onto U. The transform b' expands on higher types by b'(x) = { (b'(x 1 ), ... , b'(xn )) : (x 1 , ... , Xn) E x } ; thus b'(x) E ut � no ( t} whenever X E D+ t . Take notice that b'(D+ t ) = ut . Thus b' = b'na defines a 1 - 1 correspondence between D+ 00 and U00 • Sim ilarly the map E = 0Eb defined on E by c {b) = 0 and c {b) = {b} for all b E E is an internal bijection from E+ onto V expanding on higher types t as above, so that we get a 1 - 1 correspondence between E+ 00 and V00 • Now, given a parameter-free £00-formula 1/J(xt ) with xt as the only free variable, one easily defines another £00-formula, denoted by 1/Jv (x, �o ( t) ) , containing D and some sets ut as parameters - this is symbolized by the subscript D since the sets ut involved are derivates of D - so that 2![D+] t= 1/J(xt ) iff 2![D] t= 1/Jv (a, b'(xt)), for any xt E D+ t · (For instance 1 er 3 yt ... yt .. . 1. s changed to 3 TJo ( t } E ut . .. TJo ( t} every quant"fi th1' s s hows how the sets ut appear as parameters.) Then we have 2![D] t= 1/Jv {a, b'(xt )) iff 23 [E) t= 'l/JE (b, p6 ( t) (b'(xt ))) for any type t and every xt E D+ t by Lemma 7.2. 14. In the last formula, one can easily verify that p6 (o(xt )) = E(yt) where yt = P+ t(xt ) . We conclude that the final statement, 23 [E) t= 1/JE (b, E(yt)) , is equivalent to 23 [E+ ) t= 1/J(yt), similarly to the first step of this argument in the preceding paragraph. Thus the equivalence 2![D+) t= 'l/J(xt ) iff 23 [E+) t= 'l/J(P+t(xt )) holds for any type t and any xt E D+ t . The case of formulas with more than one 0 ( Theorem 7.2. 12) variable does not differ much. •••
-
7 .2f Generic isomorphisms
To prove Theorem 7.2.7, define, in IH, the p. o. set IP = IP.c {2!, 23) according to Definition 7.2.10. Then define the set and P = P.c (2!, 23) = {.P : p E IP} , where p = (p, N) E IH ' 0, as in the proof of Theorem 7.2. 1. Note that P belongs to IH, satisfies P � IH ' 0, and is standard size distributive in IH together with IP by Lemma 7.2. 1 1, and hence can be used as a forcing notion. It follows from the countability of IH that there exists a set G � P, P generic over IH. Then IH[G) t= HST by Theorem 7. 1.20 and it is true in IH[G) that there exists a set G3 � IP, IP-generic over IH. Define, in IH[G), F = U G3. Then F is a bijection of A onto B by Theorem 7 .2. 12 and ordinary forcing arguments, as in the proof of Theorem 7.2.1. Then, in IH[G], F turns out to be a union of compatible conditions in IP, thus it preserves the truth of £-sentences, in particular, all atomic sentences. We conclude that the map F is an isomorphism of 2! onto 23 in IH [ G] . 0 (Theorem 7.2. 7)
7.3 Consistency of the isomorphism property
279
7.3 Consistency of the isomorphism property
Let "' be a cardinal in the ZFC universe. In model theoretic nonstandard analysis, a nonstandard model is said to satisfy the K--isomorphism property, IP in brief, iff whenever £ is a first-order language containing < "' sym bols, any two internally presented elementarily equivalent £-structures are isomorphic. It is known that even with "' = N 1 IP implies several strong conse quences inavailable in the frameworks of ordinary postulates of nonstandard analysis, for instance the existence of a set of infinite Loeb outer measure which intersects every set of finite Loeb measure by a set of Loeb measure 0, the theorem that any two infinite internal sets have the same external cardinality, etc . (See some references in comments to this Chapter.) HST admits the following general cardinal -free formulation of IP : l'i.
l'i.
Isomorphism Property: If £ is a first-order language of standard size then any
two internally presented elementarily equivalent £-structures are isomor phic.
In particular Isomorphism Property implies in HST that any two infinite in ternal sets are externally equinumerous. {Indeed take the empty language as £. Any two infinite sets are elementarily equivalent if equality is the only atomic symbol.) It follows by Theorem 5.5.8 that Isomorphism Property fails in ll.. [ D), and hence its negation is consistent with HST . The aim of this section is to prove the following: Theorem 7.3.1. Isomorphism Property itself is consistent with HST .
It follows that such an important technical tool of "nonstandard" mathemat ics as the isomorphism property can be adequately developed in the context of the nonstandard set theory HST . Theorem 7 .3. 1 is a consequence of the following more concrete theorem; the derivation of the former from the latter is the same as in § 7 .2a (the derivation of Corollary 7.2.2 from Theorem 7.2. 1). Theorem 7.3.2. Let 1H be a countable model of HST satisfying 7. 1.1. Suppose that "IH = ll.. [ D)" is true in IH. There exists a generic extension IH[G) t= HST where Isomorphism Property holds.
The forcing notion we employ to prove Theorem 7.3.2 will be a product, with *-finite internal support, of more elementary forcing notions, of the kind introduced by Definition 7.2. 10, each of which forces a generic isomorphism between a pair of internally presented elementarily equivalent structures of a language of standard size. It is extremely important that the extension will not contain pairs of this form other than those which already exist in IH, the ground model - this enables us to use product rather than iterated forcing.
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7 Forcing extensions of the nonstandard universe
7.3a The product forcing notion
We argue in IH, that is essentially in
HST . { IH is a model satisfying the conditions of Theorem 7.3.2.)
Definition 7.3.3. Fo r any *-cardinal 11-, let L(�) E 0 be the first-order lan guage { so}o < ��: where s0 is an n-ary relational symbol whenever a = A+n < � and A is a limit ordinal or 0. We shall consider a truncated standard size language £(�) = { S o }o< K A st o Let Ind be the class of all tuples of the form i = (w, �, A, 18) , called indices, such that w e *N, � is a *-cardinal, and A, 18 are internal L(�) structures. Obviously Ind � 0 . Suppose that i = (w, �, A, 18) E Ind. We set Wi = w, "-i = 11-, Ai = A, 18i = 18, and Li = L{"-i) . Then Li = L(�i ) = {so}o < "i is an internal language and Li = .C{�i ) = {so}o < "i A st o is a standard size language. Let 2li and �i denote the cor responding truncated forms of Ai and 18i; they are internally presented .Ci structures. Put IPi = IPe,i {2li , � i) : a p. o. set ordered as in 7 . 2 . 10 . 0 ·
The next definition introduces the forcing IP in the form of a product of all p. o. sets IP i, i E Ind, with *-finite support. Thus IP will consist of internal functions 1r such that dom 1r is a *-finite {internal) subset of Ind. In this case we define l1rl = dom 1r . Definition 7.3.4. IP i s the collection {clearly a proper class) of all internal functions 1r such that l1rl � Ind is an *-finite (internal) set, l1rl i:. 0, and 1r{i) E IP i for each i E l1rl . Define 1r � p (1r is stronger than p) iff I P I � l1rl and 1r(i) � p(i) {in IPi ) for all i E I P I · We set IPc = { 1r E IP : l1rl � C} for any C � Ind. 0
Obviously the classes Ind � 0 and IP � 0 are e-definable in 0 . We observe that if the Li-structures 2li and �i are not elementarily equivalent then IPi is empty; thus in this case i ¢ l1rl for all 1r E IP . The parameter w = Wi does not actively participate in the definition of IP; its role will be to make IP homogeneous enough to admit a restriction theorem.
Lemma 7.3.5. If C � Ind is an internal set then IPc is a standard size closed, and hence, assuming " IH = ll.. [ O)" , standard size distributive p. o. set. In fact the whole p. o. class IP itself is standard size distributive in some sense, but this needs a more complicated proof, which we leave as an exercise for the reader. Anyway only the result for IP c will be used. Proof. It suffices to prove that IP c is a rr� s set. {We refer to the proof of Lemma 7.2. 11.) By Collection, there is a cardinal � (in WIF ) such that �i � *" in 0 whenever i E C. Then each internal language Li = L(K-i), i E C, is a sublanguage of the language L = L{*�) = {so}o < ""K (see Definition 7.3.3).
7.3 Consistency of the isomorphism property
281
Accordingly every language Li, i E C, is a sublanguage of the language £ = {so } o{x)[G]} in IH[G) following usual patterns. To check {ii) of Theorem 7. 1 .20, consider, as in the proof of Theorem 7. 1 .20 any name f E Nms(P) such that X = f[G) is a function with dom X � S and ran X � IH. Take, in IH, an internal set C � Ind satisfying II/II � C. It follows from Theorem 7.3.10 that for any x, y e 1H and any condition 1r e P, 7r I� (x, iJ) E f implies 7r r c I� (x, iJ) E f. We conclude that /[G) = f[Gc), where Gc = GnP c. An ordinary product forcing argument shows that Gc is Pc-generic over IH. However Pc (unlike P ) is a set in IH, therefore the extension IH[Gc) satisfies Theorem 7. 1 .20, and hence X = f[Gc) e IH . Standardization follows from {ii) (see the proof of Theorem 7.1.20) . Collection. We suppose that X E Nms(P) and 4>(x, y ) is a formula with parameters in Nms(P). Let A = � x � Nms(P) be defined in IH as in the proof of Separation . It suffices to find a set of names B e IH, B � Nms(P), such that for every a E A and every condition (a, b) . The set B0 is not yet the B we are looking for. To define B, we first of all choose an internal set C such that Co � C, l1rl � C for all 1r E P, and for any i = (w, "'' lA, 18 ) E C we have (w', "'' lA, 18) E C for all w' E ·� . Each internal correct bijection h : C ontS C generates an automorphism Hh of P, see § 7.3d. Let us prove that
B = { Hh [b) : b E B0 and h E 0 is a correct bijection C o ntS C} is a set of names satisfying the required property. (To see that B is a set note that the collection of all internal correct bijections C ontS C is an internal set simply because it can be e-defined in 0 .) Suppose that a E A, {H[a], H[b]) by Proposition 7.3.11. However 11 4> 11 � Co and llall � Co by the choice of C0 , and hence H4> coincides with 4> and H[a] = a because H r Co is the identity. It follows that p I� 4>{a, b'), where b' = H[b) E B, as required.
We finally verify Isomorphism Property in the extension. Since the models IH � IH[G) contain the same standard sets, the well founded universe WIF is also one and the same in the two models. Therefore IH[G) contains the same ordinals and cardinals as IH. Furthermore all triples of the form: language - structure - structure, to be considered in the scope of Isomorphism Property in IH[G), are already in IH. Thus let, in IH, £, be a standard size first-order language containing "' symbols in IH ("' being a cardinal in IH) , and 2l, � be a pair of internally presented elementarily equivalent £-structures in IH. Finally, let us prove that 2l is isomorphic to � in IH[G) .
We argue in IH .
It can be assumed that £ = { s0 } o < tt, where s0 is an n-ary relational symbol whenever a = A + n < "' and A is a limit ordinal or 0. Then L = *£ E S is an internal {even standard) language equal to L(*K-) in the sense of Definition 7.3.3. Moreover we can identify s0 (a sybmol in £) with sb (a symbol in L(*K-) ) for any ordinal a < K-, and hence identify £, with the truncated language £(*"-) (see Definition 7.3.3). Accordingly we can consider 2l, � as .C{*K-)-structures. Now it follows from Corollary 1.3. 13 (ii) (in IH) that there exist internal L(*K-)-structures lA and 18 such that 2l, � are equal to the corresponding truncated substructures of lA, 18. Thus i = (0, *K-, lA, 18) belongs to Ind and .C = Li , 2l = 2li , � = �i, and finally P .c {21, �) = P .ci {2li , � i ) = P i .
We argue in the ZFC universe.
Note that the set Gi = {7ri : 1r E G A i E l1rl } belongs to IH[G). {Indeed, since Pi is a set in IH, a name for Gi can be defined in IH as the set of all pairs of the form (1r, p) , where 1r = { (i, p)} E P .) An ordinary product forcing argument shows that Gi is Pi-generic over IH in IH[G). But then the structures 2l and � are isomorphic in IH[Gi) (see the proof of Theorem 7.2.7 in § 7.2f). It follows that 2l, � are isomorphic in IH[G), a larger model. 0 0
{Theorems 7.3.2, 7.3. 1 )
Problem 7 3 13 Suppose that a E *N ' N . Does there exist a generic ex tension of IL[D) in which all nonstandard *-integers in U = U ne rN [O, na) are equinumerous to each other and all *-integers in *N ' U are equinumerous to each other but not equinumerous to those in the first class ? o .
.
.
7.3 Consistency of the isomorphism property
287
Historical and other notes to Chapter 7
Section 7 .1. Forcing in a non-well-founded environment has been occasion ally studied in several papers, for instance [Bof 72, Tz ** , Mat 01). Yet the version applied in this book is close to the ordinary ZFC forcing because the ill-founded "kernel" 0 of the HST universe does not change in generic extensions. Some details {including Stan d ardi zati on ) need some effort to be settled, of course. Section 7.2. See [CK 92, Theorem 5.1. 13) on isomorphism between satu rated elementarily equivalent structures in model theory and [CK 92, 5.1.11) on a back-and-forth argument in this context. The forcing IP e,{2l, 23) {Defini tion 7.2.10 above) , which induces a generic isomorphism between elementarily equivalent structures, was introduced in [KanR 97, KanR 97a). Section 7.3. Isomorphism property IP was introduced by Henson [Hen 74). Studies carried out in the 1990s (see, for instance, [Jin 92, Jin 92a, Jin 96, Jin 99, Jin ** , JinS 94, JinK 93, Schm 95), also Ross [Ross 90]} demonstrate that IP implies several strong consequences inavailable in the framework of ordinary postulates of nonstandard analysis, for instance the existence of a set of infinite Loeb outer measure which intersects every set of finite Loeb measure by a set of Loeb measure zero, the theorem that any two infinite internal sets have the same external cardinality, and many more. Typical consequences of IP for different cardinals "' can be easily converted to consequences of Isomorphism Property in HST . Isomorphism Property as a hypothesis in HST and the proof of its consis tency with HST {Theorem 7.3.1 above) appeared in [KanR 97a). 1 5 Problem 7.3.13: compare with Theorem 8 in [Mil 90]. We close this Chapter with two more problems. Problem 7.3.14. Find other significant properties on external universe whose consistency or independence can be proved by forcing. Possible can didates are several hypotheses of the existence of generic sets studied by Di Nasso and Hrbacek [DiNH 03] (see also [Jin 99, Jin **]) in the frameworks of model-theoretic nonstandard analysis. Another possible group consists of questions of the type considered by Miller [Mil 90). 0 Problem 7.3. 15. Our forcing set-up includes the principle that no new in ternal sets can be added, and for good reasons. Nevertheless, if S[G) is a usual, ZFC-like generic extension of S, a standard core of a model IH t= HST, is there any way to naturally define a model IH[G) t= HST which is a standard core extension of S[G) and simultaneously a generic extension of IH ? 0 K
K
15 The question how to accomodate advanced nonstandard tools like the isomor
phism property in a reasonable nonstandard set theory was discussed in the course of a meeting between H. J. Keisler and one of the authors (V. Kanovei, during his visit to Madison in December 1994).
8 Other nonstand ard theories
The "Hrbacek paradox" {Theorem 1.3.9) can be viewed as the statement of inconsistency of the conjunction of the four following axioms, over a weak nonstandard theory: - Collection ;
- either of the axioms of Choice and Power Set; - standard size Saturation; - Standardization .
Any solution of the paradox means that (at least) one of the axioms has to be abandoned or essentially weakened to a form compatible with the other ones. The theory HST, the main topic of this book, sacrifices both Choice and Power Set {keeping either of them in a standard size form, and fully in suitable partially saturated universes). Other solutions are possible: we can keep any three of the four axioms and a partial form of the fourth one, which leads to theories based on different views of the nonstandard universe. This Chapter contains a brief exposition of the theories NST, KST, *ZFC obtained this way. We begin in § 8.1 with Kawai's theory KST which keeps Collection, Power Set, and Choice but reduces Standardization to a form com patible with the assumption that S and 0 are sets rather than proper classes, and NST, another of Hrbacek's theories {§ 8.2) which abandons Collection in favour of Power Set and Choice. Di Nasso 's theory *ZFC, designed to avoid the Hrbacek paradox by reducing the amount of Saturation available to cardi nals €-definable in WIF, is considered in § 8.4a. These theories offer adequate tools to develop nonstandard mathematics, and {especially, KST) have ad vantages relative to HST in some details. All of them are conservative, but, unlike HST, not "realistic" {in the sense of Definition 4.1.8) extensions of ZFC, hence, they hardly can be anything more than syntactical deduction schemes with respect to the "standard" universe of ZFC. Some other nonstandard theories will be considered, most notably, the Ballard - Hrbacek system based on Boffa's non-well-founded set theory. The connection between well-founded and standard sets, on which the scheme " WIF --4 0 [ in IH)" is based in HST (see § 1.2a) , will not be valid any longer for most of the theories considered in this Chapter, although partial schemes of this kind will usually work.
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8 . 1 Nonstandard set theory of Kawai
Unlike HST, Kawai's nonstandard set theory KST describes the class 0 of internal sets as a universe satisfying internal set theory 1ST rather than bounded set theory BST. This does not allow to use Hrbacek's definition of internal sets as elements of standard sets, therefore, in Kawai's system int x ("x is internal") is an atomic predicate. The theory KST contains both Power Set, Choice, and Collection, actually, all of ZFC except for Regularity, so that it is well equipped technically as a nonstandard theory. This is a solution of the "Hrbacek paradox" at the cost of Standardization: this axiom is weakened to a form compatible with the assumption that S is a set rather than a proper class (as in most of the nonstandard set theories). Metamathematically, KST is still a conservative extension of ZFC, but not a "realistic" nonstandard theory. Theories NST and *ZFC, which solve the paradox by weakening, resp., Collection and Saturation, will be presented below.
S.la The axioms of Kawai's t heory Thus, KST is a theory in £e, st , int , the language containing the membership E and the unary predicates , st, int as atomic predicates. The list of axioms of KST includes:
1) all axioms of § l.lb {the first group of HST axioms), with the schemata of Separation, Collection, Replacement in the language £e, st , int ; 2) Power Set and Choice in their ordinary ZFC forms in the €-language, as in § l.lh; 3) Transfer, Transitivity of 0 , Regularity over 0, ZFCst of § l.lc {the second group of HST axioms), however, int is now an atomic predicate of the language rather than the formula 3st y (x E y) as in HST ; and three more axioms: Set-existence of 0 : 0 = { x : int x } is a set and S � 0 ; Restricted Standardization: ys t S V X � S 3s t y (X n S = Y n S) ; Strong Saturation: if !C � 0 is a n-closed set of S-size, i. e., a set of the form y = {f(x) : X E S} , then n !C # 0 . The first axiom implies that S is also a set by Separation. This immedi ately makes the HST Standardization inconsistent with KST, yet essentially Restricted Standardization expresses the same property because in HST any way every set X � S satisfies X � S for a standard S by {3) of Exer cise 1 . 1 . 1 1 . On the other hand, Kawai's theory KST admits a bigger amount of Saturation than HST : indeed, any set of standard size is obviously a set of S-size as well, but not vice versa.
8.1 Nonstandard set theory of Kawai 291 Exercise 8.1.1. Show that if � is an axiom of 1ST then �int is a theorem of KST, thus ( D ; E, st) is an interpretation of 1ST in KST. Yet KST does not prove that D is formally a model of 1ST : this fact follows from Theorem 8. 1.5 by the same argument as in Exercise 1.5.17. 0 Define, in KST, .s = card S and i = card D ; both sets (as well as any other set) are well-orderable, hence, have cardinals in KST . Similarly to HST, the whole universe of sets is postulated in KST to be a ZFC-like world over D as the collection of "atoms" , but, unlike Hrbacek 's theory, KST sees D (as well as S � D ) as sets rather than proper classes. This property, and the weakened Standardization , is why the Hrbacek paradox does not work in KST despite the presence of Collection .
Exercise 8.1.2 ( KST ) . Show that the class WIF of well-founded sets is tran sitive and �-complete {that is, y � x E WIF ==> y E WIF ) , and an interpreta 0 tion of ZFC . (Compare with Theorem 1.1.9 in HST !) In KST, *-methods can be developed to a great extent.
Definition 8.1.3. A set x E S is condensable if there exists a transitive set X E S containing x and a map y -+ y defined on X n S such that y = { z : z E y n S} for all y E X n S. In particular, x is defined in this case, and, as E t S is a well-founded relation {Theorem 1.1.9{i) remains valid in KST ), x is a well-founded set independent of the choice of X. 0 Put WIFfeas = { x : x E S} Ueasible well-founded sets). Collection makes all standard sets condensable in KST (as well as in HST, where x = "'(x) for any x E WIF, Exercise 1.1.8, and WIFfeas = WIF ).
Exercise 8.1.4 (KST). {1) Prove that WIFfeas is a transitive and �-comp lete subclass of WIF, and the map x � x is 1 - 1, and hence for any u E WIFfeas there is a unique set x E S, denoted by *u, of course, such that u = x. Thus u � *u is an €-isomorphism of WIFfeas onto S, so that WIFfeas is an interpretation of ZFC. {2) Prove that WIFfeas is a set, moreover, WIFfeas = Vs , where .s = card S , yet KST does not prove that WIFfeas is a model of ZFC, for otherwise KST proves Consis ZFC, which is impossible by Theorem 8. 1.5 below. (3) Prove that V� [D) is a transitive set in KST for any {well-founded) ordinal �, and every set X belongs to u� eOrd v� [D). (Compare with § 1.5b.) 0 It follows that the theory KST is strong enough to develop the *-approach to nonstandard mathematics practically to the same extent and in the same way as in HST, with the only difference that the domain WIFfeas of * is a part (possibly proper) of the whole class of well-founded sets. This approach can be called the scheme ''WIFfeas � D [ in IH) ".
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8 Other nonstandard theories
8.1 b Metamathematical properties Recall that a nonstandard theory � in the st-E-language is a conservative standard core extension of ZFC in the case when any €-sentence � is a theorem of ZFC iff the relativization �st is a theorem of �-
Theorem 8.1.5. The theory KST is a conservative, and hence equiconsis tent standard core extension of ZFC .
Proof. Our plan is to define an interpretation of KST in ZFC19, a standard theory studied in §§ 1.5f, 4.4c. The interpretation will be of a kind sufficient to derive the theorem. We argue in ZFC19 . Let V = ViJ . Let *v = ( "'V ; "'E , "'st) be a 19+ -saturated interpretation of 1ST plus S-Size Choi ce in ZFC19, defined in § 4.4d {for 'Y = 19 + ), with an elementary standard core embedding * : V -+ *V. {Note that *V = V"Y , * is e0"Y, and *E is E"Y in the notation of § 4.4d.) To prove the theorem we define a "superstructure" over *V that interprets KST. To avoid unnecessary compli cations, assume that all elements of *V have one and the same von Neumann rank in the ZFC19 universe. (Otherwise choose an ordinal "' with *V � V" , replace each x E *V by (x, "-), and change *E and "'s t accordingly.) Define, by induction on � < "( , a set P� , so that Po = *V and (I) P 1 is the set of all sets X � Po different from any set of the form x *E = {y E *V : y � x} , where x E "'V ; {II) if � � 2 then P� is the set of all sets X � P < � = U71 . is a transitive and �-complete set, satisfying IH >. n WIF = WIFw +>. and IH >. n 0 = 18, and an interpretation of NST. If A � w x w then Hierarchy Existence is true in IH >. . Once again, by the interpretability claim we here mean that for any axiom � of NST, it is a theorem of KST that, for any limit ordinal A, � is true in IH >. . This is weaker than to claim that IH >. is a model of NST .
Proof (Lemma) . Recall that 18 n WIF = IHIF (Exercise 1.2.17 remains true in KST ) , which is the same as V0 • As each next level V�+l [U] adds subsets of V� [U] , we have the equality IH � n WIF = WIFw+� by induction on � . In particular, we have IH >. n Ord = w + A .
The transitivity and �-completeness easily follows from the definition. To prove 0 n IH >. � 18 note that, similarly to Lemma 1.5.8, if U E 18 and � E Ord then any internal x E VdU) belongs to &int n (U) for some n E rN , hence, belongs to 18. This argument also justifies Full Boundedness in IH >. . (We leave details here as an exercise for the reader.) Validation of axioms of ZC (minus Regularity) and the axiom of Transitive Hulls is a rather routine exercise, for instance, Extensionality holds in any transitive class, Separation and Choice hold in any �-complete class of the
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8 Other nonstandard theories
KST universe, and the axiom of Power Set holds in IH>. because if X E v� [U] (U transitive) then 9'{x} E V�+ 1 [U). To verify Hierarchy Existence in IH>. under the assumption A � w x w, note that any ordinal � E IH>. satisfies � < w + A by the above, hence, if A � w x w then actually � < A . Consider the axioms of NST related to standard and internal sets. Axioms of ZFC8\ Transfer, Transitivity of 0 , Regularity over 0 easily follow from the construction and the fact that IH>. n 0 = 18, an interpretation of BST .
Standardization. Suppose that X E IH>., X � S. We have got Full Bounded ness already, hence, there is a set B E 18 with X � B. There is a standard set s with B E s. Then, as usual, S = U s E S and X � S, hence, by Restricted Standardization of KST, there is a standard set Y with X = Y n S . Saturation: follows from Strong Saturation of KST because any set internal
in the IH>.-Sense belongs to 18, and, on the other hand, as 18 is transitive, all elements of sets in 18 belong to 18 . Standard Size Choice and Dependent Choice: this is entailed by the full Choice axiom of NST . 0 (Lemma )
In view of the lemma, it now suffices to provide additional sentences of the theories {A), (B), (C) to be true in IH>., by an appropriate choice of A. Version 1 : 'Y = w. Then, by the lemma, the universe of well-founded sets in IH>. is Vw+w , and hence IH>. is a model of the theory (A} by the above. Version 2: 'Y = .s, where .s = card WIFfe as , see § 8.la. The universe of well founded sets in IH5 is V5 = WIFfe as , which easily implies that IH6 is a model of the theory (B). Version 2: A = .s + . Such an assignment leads to WIF n IH5+ = Vs+ , which is strictly bigger than WIFfeas = V6 , hence, 1Hs+ is a model of (C). Finally consider the theory (D). First of all, as KST is a conservative extension of ZFC, the theory KSTI = KST + "it is true in S that there is a strongly inaccessible cardinal" is a conservative extension of ZFCI, hence, it suffices to prove the consistency of (D) in KSTI. Arguing in KSTI, let "' E WIFfeas be such that "'"'- is a strongly inaccessible cardinal in S. Then "' itself is an inaccessible (well-founded} cardinal in WIF, hence, V = VK is a {transitive and �-complete) E-model (not merely an interpretation, as 0 ) of ZFC.
Exercise 8.2.13. Prove that *V (i. e., the structure ( •v ; E, st) ) is a model of 1ST, while "'V8 = Uv e v •v {the bounded part of "'V } is a model of BST and a transitive subset of *V. (Hint. To prove Transfer for •va argue as in the proof of Transfer for 18 in Theorem 3.4.5(i).) o Similarly to the above, define IH K ["'V8) = U u e •va VK [V).
Exercise 8.2.14. Prove that IH K ["'V8) is a model of NST. (Hint: accomodate the arguments used above w. r. t. theories {A} - (B).) o
8.2 "Nonstandard set theory" of Hrba.Cek
301
This reduces our task to the axiom of Standard Size Collection: this is where the inaccessibility of "' will be most essential. As we have SniHK [*VB] = S n *VB = {"'v : v E V} , it suffices to check that any set X � IHK ["'VB] of cardinality card X < "' (strictly) can be covered by a set in IHK [*VB]. For any x E X there is a set Ux E *VB and an ordinal �x < "' such that x E V�:r [Ux]· Moreover, by definition, there is a set Vx E V such that Ux � "'vx , hence, X E v�:ll [ "'vx ]· As "' is inaccessible, � = SUPx e X �X is an ordinal < "' ' and similarly, v = Ux e X Vx E v. It follows that X � v� [*v ) , but the set v� [*v] easily belongs to IHK [*VB] . 0 (Theorem 8.2.10)
8.2d Remarks and exercises Our strategy was to derive metamathematical properties of NST and its versions by interpretations in Kawai's theory KST. Yet the result can be obtained directly.
Exercise 8.2.15. Arguing in ZFen, consider the following amendments in the proof of Theorem 8. 1.5. Let *ViJ = ( *V ; "'E , "'st) now be a n-saturated standard core extension of V = ViJ . (To obtain such a model apply the quotient power chain construction of § 4.3d with 'Y = n (the length of the chain) and ( V ; E) as the initial structure.) Define P� as in the proof of Theorem 8. 1.5. Choose a limit ordinal 'Y and put P = U� 3 ! x J(cp, x)) A \;/ cp \;/ x (J(cp, x ) => SeqtN (cp) ) , where SeqrN (cp) means that cp is an N-sequence, i. e., a function with dom cp = N. Assuming this, let, for any cp with SeqrN (cp), J(cp) be that unique x which satisfies J (cp, x) .
A/pha-2: If f is a function defined on a set A and cp , 1/J : N -+ A then J (cp) = J( 'l/J ) implies J(f o cp) = J (f o 1/J ) . A/pha-3: J (cm ) = m for any m, where cm ( k) A/pha-4 : If
=
m for all k E N .
19(n ) = { cp(k), 1/J(k) } for all k then J(19) = { J(cp}, J('l/J )} .
A/pha-5 : Generally, J(cp) = {J('l/J ) : 1/J (k) E cp(k) for all n } for any cp . A/pha-6: J ( id) ¢ N, where id ( k) = k for all k .
Define *x = J(cx) for any set x (where cx (k) = x for all k in N). Sets of the form *x can be called "standard" while sets of the form J(cp), cp being an N-sequence, "internal" . Let S, 0 be the classes of all "standard" , resp., "internal" sets. Finally, define a = J ( id) : a very important set, see below. Direct arguments in ZFC[J] are quite special and do not follow any or dinary "nonstandard" intuition; even typical basic facts like J(cp) = J ('ljJ) whenever cp(k) = 1/J (k) for almost all k need some tricks. Still there is a sequence of rather simple claims:
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8 Other nonstandard theories
Exercise 8.3.9. Prove, in ZFC[J), the following: {1) If tJ {k) = (rp(k), 1/J(k)), V k, then J(tJ) = (J(rp), J('l/J)). {Recall that ordered pairs are formally defined by (a, b) = { {a}, {a, b} }. ) {2) If rp : N -+ X then J(rp) E *X, conversely, any x E *X has the form J(rp) for some rp : N -+ X . {3) If X � Y then *X � *Y, further, *(X U Y) = *X U *Y, the same for n, \, 6 (symmetric difference), x {Cartesian product). {4) If X is finite then *X = {*x : x E X} . {5) The class 0 is transitive and equal to {y : 3s t x (y E x)} . {6) If f : A -+ B then *f is a function ::4. -+ *B and *f(J(rp)) = J(f o rp) for any rp : N -+ A. In addition, if C � A then j t *C = *(! t C) . {7) If rp{k) ¢ 1/J(k) for all k then J(rp) ¢ J('l/J). Similarly, rp{k) f. 1/J(k) for all k then J (rp) f. J (1/J) - therefore, if x f. y then *x f. *y . Hints. {1) We have J(tJ ) = {J(rp'}, J('l/J')} by Alpha-4, where rp' (k) = {rp{k) } and 1/J'{k) = { 1/l {k), 'l/J {k) }. Apply Alpha-4 for rp' and 1/J'. {2) Both assertions follow immediately from Alpha-5 because *X = J(cx }, where cx {k) = X for all k. {3} By (2), a typical element of *X is J(rp), where rp : N -+ X, but then J (rp) E *Y as well. Similar arguments validate the rest of the claim. {4) Note that {*x} = *{x} {generally, {*x, *y} = *{x, y} ) by Alpha-4 . Then apply (3) for U by induction on card X . (5) Suppose that y = J{rp). Let x = ran rp = { rp(k) : k E N } , so that rp : N -+ x. Then y E *x by {2). The converse is similar. The transitivity immediately follows from Alpha-5. {6} We have *f � � x *B by (3). If a E �, then, by Alpha-5, a = J(rp), where rp : N -+ A. Put tJ (k) = (rp(k), /(rp(k))), then J(tJ) = (J(rp), J(forp)) E j by {1), {2), hence, dom j = �. Now, let (a, b) and (a', b') are two typical elements of *J, so that, as above, a = J(rp), a' = J(rp'), b = J('l/J), b' = J('l/J'), where rp, rp' : N -+ A and 1/J = f o rp, 1/J' = f o rp'. If a = a' then we have b = b' by Alpha-2. Thus j is a map � -+ *B. The equality *f(J(rp)) = J(forp) has actually been established. The additional claim: both j and *(/ t C) are functions with domains � and *C � �, and *(/ t C) � j . (7) Let tJ (k) = { rp{k) }. Then we have tJ (k) ' 1/J(k) = tJ {k), V k, hence, by {3), J{tJ) ' J('l/J ) J(tJ ), in other words, J(tJ) ' J('l/J) = 0. However J(tJ) = { J(rp)} by Alpha-4. The other claim is analogous. 0 =
Taking rp = id and f = rp in {6), we obtain *rp(o:) = J(rp) for any N sequence rp. This gives a much more meaningful form to the whole structure of the universe of ZFC[J) : J-extensions turn out to be just values of the * extended functions on a nonstandard number a ( a E *N by {2) and ¢ N by Alpha-6). We can now reformulate all axioms of ZFC[J), for instance, Alpha-3 takes the form:
8.3 Non-well-founded set theories
309
- if 'l9 {n) = { cp{k), 1/J{k) } for all k then � (a) = { *cp (a) , 'f; (a) } . But to understand what 'VJ is, we have to return to J-formulations. With heavy abuse of notation, [BenDN 03) gave all axioms prima facie in terms of a , simply dropping stars, e. g., Alpha-3 takes the form: - if 'l9 {n) = { cp{k), 1/J{k) } for all k then 'l9 {a) = { cp(a), 1/J (a) } , followed by a comment that cp(a) should be understood as the value, on a, of an extended function cp. This version, called ZFC[a] , can be rigorously understood only on the base of ZFC[J] or something like that. To gain even more clarity, let o/1' = {X � N : a E *X} .
Exercise 8.3.10. Prove that o/1' is an ultrafilter on N, containing all cofinite subsets of N. (Hint. That o/1' is an ultrafilter easily follows from (3) . If X = {0, 1, 2, ... , n} then X = *X by {4) of Exercise 8.3.9 and Alpha-3, hence, if X E o/1' then a = n for some n E X, contradiction with Alpha-6.) 0 Theorem 8.3.11 (Los Theorem, ZFC[J]) . If a(x1 , , xn) is an €-formula •••
and cp1 ,
••.
, 'Pn are N-sequences then
(t) a(*cp 1 (a), ... , *'Pn (a)) is true in 0 iff { k : a{rp l {k) , ... , rt'n {k)} } E o/1' . If, moreover, a is a bounded formula (see § 1.5a) then
(t) a(*cp 1 {a), ... , *'Pn (a)) iff { k : a{cpl {k) , ... , 'Pn (k))} E o/1' .
Proof. The "moreover" part follows because bounded formulas are absolute for any transitive class. { 0 is transitive by {5) of Exercise 8.3.9.) The main part is proved by induction on the complexity of a. Let a be x E y and cp, 1/J be N-sequences. Consider the set U = {k : rp{k) E 1/J{k) } . Suppose that a E *U. Define rp', 1/J' so that they coincide with resp. cp , 1/J on U while cp' {k) = 0 and 1/J' {k) = 1 = { 0 } on N ' U, so that rp' {k) E 1/J' {k) for all k. Then *(rp' ) (a) E *(1/J')(a) by Alpha-5. On the other hand, as a E *U, we have *(rp')(a) = *(cp' t U) (a) = *(cp t U) (a) = *cp(a) by the additional claim in {6) of Exercise 8.3.9, and similarly *(1/J'){a) = 'VJ(a), hence, *cp(a) E 'VJ(a). If a ¢ *U then *cp(a) ¢ 'VJ(a) by analogous arguments, but {7) of Exercise 8.3.9 is used instead of Alpha-5. Formula x = y is treated similarly. As usual, the inductive steps for A and ..., are rather easy, thus, we can con centrate on 3 . Suppose that U = { k : 3 x a(x, 'Pl {k) , ... , 'Pn {k))} belongs to o/1' . By Choice, we obtain an N-sequence rp such that a{cp{k) , rt'l {k) , .. . , 'Pn (k)) for all k E U. Then a{*cp(a), *cp 1 {a) , ... , *'Pn(a)) is true in 0 by the in ductive hypothesis, therefore, as x = *cp (a) = J{rp) E 0, the formula 3 x a(x, *cp 1 {a), .. . , *rpn (a)) is true in 0, as required. Conversely, suppose that a(x, *cp1 {a), ... , *'Pn {a)) is true in 0 for some x = *cp(a) E D. Then the set W = {k : a{rp{k) , cpt {k), ... , cpn{k))} belongs to 'PI by the inductive hypothesis. However clearly W � U. 0
310
8 Other nonstandard theories
Corollary 8.3.12. *«p(o:) = *1/J(o:) iff the set U = { k : cp{k) = 1/J(k) } belongs 0 to au , in particular, it suffices that U is co.finite. The same for E . Corollary 8.3.13 (*- Transfer, ZFC[J] ) . H a (x1 , ... , xn } , is an E-formula and x1 , . . . , X n are any sets then {t) a ( *x1 , .. . , *xn) is true in 0 iff a (x1 , ... , xn) . If, moreover, a is a bounded formula then 0 (t) a {*x 1 , .. . , *xn) a (x1 , .. . , Xn) . It follows that 0 is just (isomorphic to) the ultrapower Ult%' {V) of the whole set universe V of ZFC[J) ! This brings ZFC[J) back on the track of ordinary nonstandard methods, with the following special features: 1 o . * is an elementary embedding (in the sense of the E-language) of the whole set universe V into the class 0 of all "internal" sets, and which is the transitive closure of the range ran * in the same universe V . 2° . 0 is a Gordon class, in the sense that there is a E *N ' N such that 0 consists of all sets of the form *f(o:), where f is a function defined on N. Countable Saturation comes for free: Proposition 8.3.14 (ZFC[J]) . The class 0 is countably saturated. Proof. Suppose that Xn = *'Pn (o:) E 0 are nonempty sets which form a f. i. p. family. We can assume that Xn+ l � Xn for all n. For any n, the set Un = { k : 'Pn (k) i:. 0} belongs to au by Theorem 8.3.1 1 , hence, we can assume that 'Pn (k) i:. 0 for all k, for if not redefine 'Pn outside of Un and use Corollary 8.3.12. Similarly, it can be assumed that 'Pn+l (k) � 'Pn (k) for all n, k. Choose any 'l9 (k) E cpk (k). In our assumptions, 'l9 (k) E cpn {k) for all k ;::: n, thus, x = "i9( a ) belongs to any Xn = *'Pn ( a ) by Corollary 8.3. 12. 0
8.3e Interpretation of Alpha theory in ZFBC Theorem 8.3.15. There is an interpretation of ZFC[J) in ZFBC with the same set universe. Therefore ZFC[J) is wt-core interpretable in ZFGC by Theorem 8.3.3, and hence ZFC[J) is a conservative wt-core extension of ZFC . Proof. Arguing in ZFBC, fix a nonprincipal (i. e., containing all cofinite subsets of N } ultrafilter au on N. The ultrapower Ult %' (V) = ( ·v ; •E ) of the whole universe V is then an extensional structure, hence, by Lemma 8.3.6, there is a transitive class 0 and an isomorphism 1r : ( ·v ; •E ) ontS ( 0 ; E ) . The superposition x � *x = 1r { •x) of 1r and the canonical embedding x � •x of V into ·v is the an elementary embedding of (V ; E) into ( 0 ; E) . To define J, let c E ·v be the au -class of id {recall that id{k) = k for all k ) . Let a = 1r(c) and put J(cp) = *«p(o:) for any map cp defined on N. We leave it as a (difficult !) exercise for the reader to prove that (V ; E, J) is an interpretation of ZFC[J) (in ZFBC). o
8.4 Miscellanea: some other theories
Problem 8.3.16. Is ZFC[J) wf-core interpretable in ZFC ?
31 1
0
Coming back to principles 1° and 2° in § 8.3d which, in a sense, char acterize the theory ZFC[J) , it is quite clear that while the former is really important for development of nonstandard analysis in ZFC[J), the latter is rather special, moreover, an easy argument shows that 2° is incompatible with Saturation for families of cardinality ;::: 2No . Therefore, it looks natural to drop 2° but add to 1° more Saturation. This leads to a theory {let us denote it by ZFCK [*D in the €-language enriched by two additional symbols, * and "'' with the following axioms: 1) all of ZFC without Regularity ( * and "' can occur in the schemata), 2) axioms saying that "' is an infinite {well-founded) cardinal while * is a map (a proper class) defined on the whole set universe V, 3) *-Tra nsfer for * as a map V -+ 0, where 0 = {y : 3 x (y E *x ) } , 4 ) Transitivity of 0 , and 5) Saturation for families !C � 0 of cardinality :$ K..
Exercise 8.3.17. Replace 5) by a stronger requirement: Saturation for well orderable families !C � 0. Why is this inconsistent ? 0 Exercise 8.3.18. Arguing in ZFBC, let "' be an infinite (well-founded) car dinal. Take V = V in Theorem 8.3.8, and let * be an elementary embedding of V into a transitive and 11-+ -saturated class *V. Show that then ( *V ; E, 11-, *) is an interpretation of ZFCK [*] . 0 8.4 Miscellanea: some other theories
We begin this section with a nonstandard set theory, due to Di Nasso, which circumvents the Hrbacek paradox by reducing Saturation to a form that still incorporates all definable cardinals. Then three "stratified" nonstandard the ories are considered: their common property is that a single "universe of discourse" is replaced by a conglomerate of universes related with each other in a certain way. Finally, a nonstandard class theory will be considered.
8.4a A theory with "definable"
Saturation
The fourth, and last solution of the "Hrbacek paradox" (see the beginning of § 8.2) is to reduce Saturation to a form compatible with Power Set + Choice + Collection + Standardization. At first glance the task does not seem to have an adequate solution. In particular, because no cardinal, chosen as the amount of Saturation postulated, can be consistently argued to fulfill all needs of nonstandard mathematics once and for all. Moreover, fixing any cardinal for this purpose is neither esthetically nor philosophically acceptable.
312
8 Other nonstandard theories
A modification not connected with any particular cardinal was suggested by Di Nasso [DiN 99). Let DNST be HST amended as follows 6 : Power Set and Choice are added, but Saturation is postulated for families whose cardinality is a cardinal €-definable in WIF. Thus, Satu ration in DNST is an axiom schema, which we call Definable Saturation, containing, for any E-formula rp (x ), an axiom, say, SATcp, saying that Saturation holds for all n closed families (of internal sets) of cardinality � 11-, where "' = "-cp is the least infinite cardinal satisfying rpvf (11-) , or No if no such cardinals exist. This does not imply Saturation for all standard size families (of any cardinality), more over, we can consistently define the least cardinal "' for which Saturation fails, but this is not an €-definition in WIF ! DNST is still a conservative standard core extension of ZFC. Indeed, it suffices to prove that each subtheory DNSTcp, where we have only SATcp instead of the whole Definable Saturation, is a conservative standard core extension of ZFC. Arguing in ZFC, define a cardinal k = "-cp as above. As HST is a "realistic" theory, there is a standard core interpretation 1H = ( 1H ; *E , *st) of HST in ZFC, together with an associated canonical €-isomorphism * : V ontS S = S( IH} , where V is the set universe of ZFC. The classes S and WIF = WIF ( IH} are €-isomorphic to each other in HST, hence, by superposition, there is an €-isomorphism, say 1r : V ontS WIF. Let "' = 1r{k), so that it is true in IH that "' is a {well-founded) cardinal. However (see "second option" on p. 253), there is a class, say, IH� � IH, satisfying HST�. It follows that IH� is a standard core interpretation of DNSTcp in ZFC, hence (see Proposition 4.1.9) DNSTcp is a conservative standard core extension of ZFC .
Exercise 8.4.1. Show that DNST is not a "realistic" nonstandard theory in the sense of Definition 4. 1 .8. (Hint. A minimal ZFC model M is not DNST extendible since, as all sets in M are €-definable in M, see Exercise 4.6.22, such an extension would be a model of the full standard size Saturation , contrary to the Hrbacek paradox.) 0 8.4b Stratified nonstandard set theories Under this title, we gathered three theories which have the common property of being focused on certain parts of the nonstandard universe rather than on the latter as a whole. We give here a rather sketchy review of the theories and refer the reader to original papers for details, in particular, regarding the proofs of their conservativity and equiconsistency with ZFC. (Reservation: Fletcher's presentation of SNST in [Fl 89) is very sketchy.) Fletcher's stratified nonstandard set theory. The theory SNST defined in [Fl 89) sees the nonstandard universe as the union of a system of internal 6 Actually Di Nasso's formalization uses * as a primary notion, while st is a definable predicate, i. e., st x iff x = •u for some well-founded u.
8.4 Miscellanea: some other theories
313
subuniverses lex and external subuniverses Ec:x, where a is a cardinal in the standard universe S {which satisfies ZFC, as usual). This system of subuniverses looks rather similar to the system of classes 0 " and ll.. [ D " ) in HST, with some minor differences, for instance, �-size Saturation rather than K-deep Saturation is postulated. Ballard's enlargement set theory. The nonstandard theory EST defined in [Bal 94) has a definite flavour of category - theoretical ideas: it essentially denies anything like a common "working" set universe, but instead postu lates a conglomerate of universes connected via embeddings so that still each universe admits an elementary embedding into another, suitably saturated, universe. The whole picture can be compared to a system of transitive classes within a universe of Boffa's set theory ZFBC, which consists of those classes which satisfy a certain version of the von Neumann- Godel - Bernays class theory NBG {Theorem 8.3.8 validates the existence of sufficiently saturated extensions.) The following citation from [Bal 94, p. 128] gives an impression of Ballard's philosophical position: "In designing the vehicle EST, I have de
liberately ignored the needs of practitioners and sought instead to decisively illustrate the full implications of this relativistic mathematical ontology. " Theories of relative standardness. Peraire ' s system RST [Per 92, Per 95) utilizes st as a binary predicate, i. e., in the form x sty, which is understood as x is standard relative to y. This is a theory of internal kind, like BST or 1ST, and its universe has some semblance to a BST universe where x st y is defined by the st- E-formula x E S[y) (see Definition 3. 1.13 ) , but the whole structure of axioms is closer to 1ST. Note that the binary predicate x st y is atomic in RST, which allows to avoid the restrictions imposed by
Theorem 6. 1.15 and consistently add Inner Standardization for any class of the form 111 = { x : x st y } . Peraire demonstrated in a number of examples that the relative standard ness gives an adequate treatment for phenomena connected with double and more complicated limits in topology and analysis. Another approach to relative standardness, related rather to BST, has recently been proposed by Hrbaeek [Hr 04, Hr **). 8 .4c
Nonstandard class theories
Hrbacek's idea, that any reasonable "standard" theory of set theoretic type admits a certain nonstandard version, was applied to the von Neumann Godel - Bernays class 7 theory NBG by Gordon in [Gor 97). This resulted 7 Recall that the common feature that distinguishes class theories from set theories (both standard and nonstandard} is that the former consider classes as primary objects, while sets are distinguished as those classes which are elements of other classes. Axiomatic systems of class theories look different from those of set the ories even in the case when the theories are very close metamathematically as e. g. ZFC and NBG .
314
8 Other nonstandard theories
in the nonstandard class theory NCT (in more advanced form, see Andreev and Gordon (AnG 2001)}, which is a standard core extension of NBG in approximately the same way as BST is a standard core extension of ZFC. The universe of NCT consists of sets and classes, both types contain ing standard (satisfying st ) and nonstandard objects, with appropriate Comprehension schemata which reflect the idea that the set universe is inter nal while classes are not necessarily internal. Internal classes are introduced as follows: if X is a standard class (i. e., st X ) and p any set then the class { x : (p, x) E X} is internal. A related Comprehension axiom postulates that any intersection of a set and an internal class is still a set, hence, any internal class X satisfying X � x for a set x is itself a set {but there are non-internal classes with this property which are not sets). As for metamathematical properties, being a standard core extension of NBG, the theory NCT is, at the same time, a class extension of BST , in the sense that the set universe of N CT satisfies BST, and conversely, any model of BST can be embedded, as the class of all sets, into a model of NCT. {The latter can be obtained by adjoining all st-E-definable subclasses of the given BST universe; our Theorem 3.2.3 plays the key role to make such an extension procedure working.) Accordingly, any theorem of NCT which speaks only about sets is a theorem of BST, so that NCT is a conserva tive class extension of BST. It follows that NCT is a conservative {hence, equiconsistent) standard core extension of ZFC. Nonstandard class theories can be expected to be useful in the treatment of those phenomena in the model theoretic version of nonstandard analysis which naturally lead to class-size objects in the frameworks of a nonstandard set theory, see, e. g., Kanovei and Reeken [KanR 99b), where a version of SMA was considered in (a simplified version of) NCT . The same goals also can be achieved in Kawai's theory KST {because S � 0 are sets in KST, hence, S-size and 0-size objects are sets in KST, so that there is no need for it), however, the use of NCT has a principal advantage here, because NCT provides what seems to be the minimal reasonable nonstandard universe containing S-size objects. It remains to briefly mention THS, theory of hyperfinite sets of Andreev and Gordon, see [AnG 2001) and especially a forthcoming paper [AnG **). This nonstandard theory (actually, a class theory rather than set theory) shares some ideas with the alternative set theory AST of Vopenka, in par ticular, its set universe is intended to consist of sets with *-finite transitive closure (in the notation of HST ). The main feature of THS is that it does not make use of standard sets. However to apply Saturation-like tools there should be a suitable notion of a "small" collection of sets - and this is achieved in THS by a careful combination of €-definitions. Metamathematically, THS turns out to be as strong as the Zermelo theory ZC .
8.4 Miscellanea: some other theories
315
Historical and other notes to Chapter 8
Section 8 . 1 . Kawai's set theory was introduced in [Kaw 83) under the name: nonstandard set theory, NST. A weaker version was proposed earlier in [Kaw 81). Theorem 8.1.5: Kawai [Kaw 83). Section 8.2. The theory NST was introduced, under the name NS 2 (ZFC), in [Hr 78) , where also its conservativity is established. A more comprehensive exposition was given in [Hr 79). Our method to prove the conservativity of NST and its versions in § 8.2c by inner models in KST is, of course, rather anachronistic: Kawai's paper [Kaw 83) was published later than Hrbacek 's works. Theorem 8.2. 10: Hrbacek [Hr 78). Exercise 8.2. 7: Hrbacek [Hr 78) and private communication. Exercises 8.2. 16, 8.2.17, 8.2.18: Kanovei and Reeken [KanR OOa). Section 8.3. See [Kun 80, III.5) on the Mostowski collapse theorem in ZFC . The Universality axiom is identified as BA1 in [A 88). See also [HrJ 98, p. 265). Note that both Universality and SuperUniversality are different from {and in fact incompatible with) another rather popular axiom which implies the existence of ill-founded sets: the antifoundation axiom, or AFA, formu lated as every graph has a unique decoration, see [A 88, Dev 98, HrJ 98). AFA describes a set universe in a sense less ill-founded than those described by Uni versality and SuperUniversality, and apparently does not lead to applications to nonstandard set theories. The content of § 8.3a - 8.3c is mainly due to Boffa [Bof 72) (regarding ZFBC in general and its relations to standard set theories) and Ballard and Hrbacek [BalH 92) (regarding applications to nonstandard analysis). The key axiom of SuperUniversality {or BAFA in [A 88]) was introduced in [Bof 72). Aczel [A 88) gives a broad reference in the history of non-well-founded set theories which in fact goes back to the times of Zermelo and Fraenkel. Di Nasso's theory ZFC[o:] first appeared in [DiN 99) with a slightly dif ferent {but equivalent) list of axioms. One of its motivations was to give rigorous treatment of a pre-Robinson attempt in nonstandard analysis, due to Schmieden and Laugwitz [SchmiedL 58) .
9 "Hyperfinite" descriptive set theory
Descriptive set theory studies those subsets of topological spaces (called
pointsets) which can be defined, by means of a list of specified operations
including, e. g., complement, countable union and intersection, projection, beginning with open sets of the space. Classical descriptive set theory (DST) considers mainly sets in Polish {that is, separable metric) spaces, this is why we shall identify it here as Polish descriptive set theory. "Hyperfinite" descriptive set theory follows this scheme in a different set ting: the construction of hierarchies begins with internal subsets of a fixed infinite internal set H as the basic sets. Note that internal subsets of an infinite internal H do not form a topology, moreover, the weakest topology where all internal sets are open, is discrete because all singletons are internal, hence, H is called rather domain than space. 1 It turns out that many questions on the nature of pointsets, considered by Polish descriptive set theory, remain meaningful in the "hyperfinite" setting, sometimes directly sometimes in a more or less revised form. Accordingly, the results obtained are sometimes similar to those of Polish descriptive set theory, sometimes just the opposite. But in general "hyperfinite" descriptive set theory is much less developed than Polish DST. As for the methods, they can be very different. The following is a very rough classification of theorems of "hyperfinite" descriptive set theory from the point of view of the methods involved: (A) results similar to "Polish" theorems and obtained by virtue of the substi tution of Saturation for completeness {or compactness) in "Polish" proofs; (B) corollaries of "Polish" theorems by means of shadow maps ; (C) results that appear stronger than their "Polish" counterparts because Saturation is in some cases stronger than completeness or compactness; (D) results based on a kind of "hyperfinite" combinatorics, including plain pigeonhole-type arguments, sometimes w. r. t. non-internal objects. 1 The domain H is sometimes taken to be a *-finite, that is hyperfinite set, which is essential for applications like Loeb measures, - this is why this direction is called 11hyperfinite" DST. However most results will be true for all infinite internal domains H.
318
9 "Hyperfinite, descriptive set theory
The content of this Chapter includes the following. We begin in Sec tion 9.1 with the basic set-up including Borel, projective, Souslin subsets of internal sets. Operations of countable character, count ably determined sets, and the related concept of shadows follow in Section 9.2. Closure properties of Borel and projective classes, based on the key shadow theorem {Theo rem 9.3.3, which shows that shadow preimages keep a Borel or projective class in both directions), are considered in Section 9.3. The next Section 9.4 is central: we present main structural theorems of "hyperfinite" descriptive set theory, including Separation, Reduction, Uniformization, sets with spe cial cross-sections, and some other theorems. Some questions related to Loeb measures {like the existence of liftings) are considered in Section 9.5. Sec tion 9.6 presents studies on "Borel cardinals" , that is, relations between Borel sets in terms of Borel injections and bijections, and "countably determined cardinals" , that is, relations between countably determined sets in terms of countably determined injections and bijections. This research line continues in Section 9.7, where we study quotients over Borel and countably deter mined equivalence relations, a topic quite typical for modern works in Polish descriptive set theory. We left aside such notable topics as some foundational issues in nonstan dard real and functional analysis, topology, and Loeb measures (except for a brief Section 9.5 not at all covering the issue) , which have some relevance to "hyperfinite" descriptive set theory. Unfortunately we have also to sacrifice our original plan to add a survey of the following topics: 1) nonstandard topologies generated by count ably determined cuts in *N (see [KL 91, Jin 01) and references in the second paper) , 2) sets with special properties related to category and Loeb measure (as in [Mil 90]), 3) completeness properties of the *-reals (see [Jin 96) and references there) , - because of the limited space available for "hyperfinite" descriptive set theory in this book. Our exposition will follow the standards of "hyperfinite" descriptive set theory in model theoretic nonstandard analysis, including its commitment to countable Saturation (see Blanket Agreement 9. 1.2) and its stress on count ably determined sets. In principle all results below (except those explicitly indicated) are true in the model theoretic setting. We tried to make the exposition as self-contained as possible within a rather restricted space, yet some degree of aquaintance with Polish descriptive set theory and, to a lesser extent, with "hyperfinite" DST the in model theoretic version will be assumed. Kechris [Kech 95) is given as a general reference in matters of Polish descriptive set theory.
9.1 Introduction to "hyperfinite" DST
319
9 . 1 Intro duction to "hyperfinite" DST
Development of "hyperfinite" descriptive set theory in HST is quite similar to the model theoretic version, yet we have to pay attention to some essential details in the beginning of this introductory Section. Then we introduce Borel, projective, and Souslin sets in internal domains.
9.la General set-up The model-theoretic version of "hyperfinite" descriptive set theory deals with a basic "standard universe" , that is a structure V in the ZFC world of sets, which models a fragment of ZFC (usually equal or weaker than ZC), and a "nonstandard universe" , usually a nonstandard type-theoretic superstructure over an elementary extension *V of V, which contains, in particular, an element *x for any x E V. In HST, we change to the scheme "WIF � 0 [ in IH)" (see § 1.2a), which proposes the "standard" structure 2 WIF and the "nonstandard extension" 0, both transitive classes in a wider external set universe IH of HST. Thus, the multitude of "nonstandard universes" , which model theorists are accustomed to in ZFC, apparently disappears - we have a uniquely defined pair of the "standard" (i. e., well-founded) universe WIF and the internal universe 0 .
Problem 9.1.1. It is a challenging problem to utilize partially saturated subuniverses, of the type considered in Chapter 6, in a way emulating the ongoing study of the multitude of "nonstandard universes" in the ZFC set universe in model theoretic nonstandard analysis, especially w. r. t. questions related to "hyperfinite" descriptive set theory. 0 Blanket agreement 9. 1 . 2. Our development of "hyperfinite" descriptive set theory is compatible with the weakest reasonable version of HST, where 1° . Saturation is reduced to countable Saturation, that is, Saturation for countable f. i. p. families of internal sets. 2°. Countable Satu ration is sufficient to prove countable Extension, that is, for every sequence { Xn } nerN of internal sets Xn there exists an internal function f with N � dom f and f (n) Xn for all n ; this is a particular case of Theorem 1.3.12. 3° . Sta ndard Size Choice is reduced to c-size Choice {Choice in the case when the domain of a choice function is a set of cardinality � c = 2No ) . 3 0 =
Any strengthening of these assumptions will be explicitly indicated. Isomorphic to the true class S of all standard sets in HST, but more convenient, in particular, WIF is transitive and �-complete while S is not. 3 Recall that �-size Saturation, generally, corresponds to 2 ��:-size Choice, see Foot note 22 on page 253, or Remark 3.3.6. 2
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This is compatible with HST as well as with any of the partially saturated theories HST" and HST� introduced in § 6.4a. In different terms, this is also compatible both with the "main" scheme " WIF � 0 [ in IH)" (i. e. , with 0 as the internal domain) and the partially saturated schemes of § 6.4h, e. g., " WIF --!.-). 0" [ in ll.. [ O " ) ] " (with 0" as the internal domain, "' being any infinite cardinal, for instance, N0 ). This is also compatible with many nonstandard theories considered in Chapter 8. And finally, 9.1.2 is compatible with the ordinary assumptions of model theoretic nonstandard analysis. In fact all theorems below, except for those few explicitly marked as "full-HST" results, are valid in the set-up of model theoretic nonstandard analysis, with countable Saturation. Fortunately neither Power Set nor 11--size Choice for cardinals K. > 2No is really of importance in "hyperfinite" descriptive set theory (as they are, generally, not important to Polish descriptive set theory, except for rather special issues). Similarly, it is customary not to assume more than countable Satu ration in "hyperfinite" descriptive set theory.
9.Ib Comments on notation The remainder of this Chapter involves a special notation which deserves a few comments. Recall that if X � 0 and A E Ord then the collection X >. of all functions f : A = {� : � < A} -+ X is a set {Theorem 1.3.14) . In particular, the following collections are sets whenever X � 0 :
X fN = all infinite sequences of elements of X, i. e., maps f : N -+ X; xn = all finite sequences s of elements of X of length lh s = n E N ; x <w = Une fN xn , all finite sequences of elements of X, note that if f E X fN and n E N then f t n E xn � x <w ;
in addition, .9f in (X) = { all finite subsets of X } is a set. If X, Y E WIF then XY, X rN , xn , x <w, &(X) are sets and belong to WIF since N E WIF and WIF is a transitive and �-complete class satisfying ZFC {Theorem 1.1.9). Typical cases below are X = N and X = 2 = {0, 1} . This leads to the following sets in WIF :
N fN N <w 2N 21 2 <w
= = = = =
all infinite sequences of natural numbers; U ke N N k , all finite sequences of natural numbers; all infinite dyadic sequences, or maps N -+ 2 = { 0, 1} ; all maps I -+ 2 (where I is a set in WIF ); U ke N 2k , all finite dyadic sequences.
A set X is (at most) countable if there is a bijection f : N ont� X. Countable sets are sets of standard size {exercise: prove it !).
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On the other hand, if H is an internal set then int2H will be systematically used to denote the {internal) set of all internal maps f : H -+ 2 . As usual, N 1 = w 1 is the least uncountable ordinal. According to § 1.2c, w 1 is the first uncountable ordinal and the least uncountable cardinal in WIF, hence, as WIF is a model of ZFC, w1 behaves in ZFC-like manner in HST . In parallel with notions and concepts of "hyperfinite" descriptive set the ory, we shall systematically consider and refer to notions and results related to Polish spaces, especially compact spaces of the form 21 where I is a count able set. A natural open base Base{21) of 21 consists of all non-empty finite intersections of sets of the form {x E 21 : x{i) = v} , i E I, v = 0, 1. By compactness any clopen Y � 21 is a finite union of sets in Base{21). For any Polish space !C , for instance !C = 21, I being a countable set, E� [ !C), IT� [ !C) , A� [ !C) (� < w1 ) and E� [ !C), IT� [ !C) , A � [ !C) { n E N ) indicate the classes of resp. Borel and projective hierarchies of subsets of !C. We define separately E8 [ !C) = rrg [ !C) = ag [ !C) = Clop( !C) (clopen subsets of !C ) for any space !C of the form 21 . H U is a given set then we put x C = U ' X, the complement of X. As C x obviously depends on U, too, this notation will be used only if it is clear from the context which basic domain U is considered.
9.1c Borel and projective sets in a nonstandard domain Similarly to Polish DST, "hyperfinite" descriptive set theory classifies sets in accordance with the complexity of their definitions or constructions from certain initial sets, the latter being now just internal sets. Borel sets over an algebra. Suppose that Jl1 � f!ii' (H) is an algebra of subsets of a given set H, that is a collection of subsets of H closed under unions and intersections (of two sets) and complements to H. The Borel hierarchy over d consists of Borel classes 4 E� {d), IT� {d), A� {d) of subsets of H defined by induction on � < Wt as follows:
IT8{d) = A8(Jl1) = E8 (Jl1 ) = d ; IT� {d) = {XC : X E E� (Jll ) } for any � < Wi J where XC = H ' X ; A� (Jll) = E� {d) n IT� {d) for each � < Wi J i. e. a set belongs to A � (Jll) iff it belongs to both E� {d) and IT� (Jll) ; E� (Jll ) (� � 1 ) consists of all countable unions of sets in U 71< � IT� {d) ; in other words, X belongs to E � {d) iff there is a sequence {Xn}ne rN of sets Xn E u7] as follows: for all x E H and s E 2 <w, r{x){s) = 1 if and only if 3 y E K (s C cpz (y)). We claim that T is a shadow in the sense of 9. 2. 12. It suffices to prove that the r-preimage r - 1 {C8 ) of any set of the form C8 = {c E 2 0, and x � y means that lx-yl is infinitesimal. A *-real x is limited iff lxl < *c for a real c E IR . By Lemma 2. 1.9, every limited *-real x is near-standard, that is there is a unique real r = 0X E IR such that x � *r, the shadow of x . Suppose that H is an internal set and J1- is an internal measure on &int (H), that is, an internal function J1- : &int (H) -+ * [0, oo ) satisfying the requirement of finite additivity: J.l-(X U Y) = J.l-(X) + Jl-(Y) - Jl-(X n Y) for all internal X, Y � H. By the above, if X E &int (H) and J.l-(X) is a limited *-real then there exists a unique real number r = 0J.1-(X) E IR such that *r � J.l-(X). If the value J.l-(X) is non-limited then we naturally put 0J.1-(X) = 00.
Exercise 9.5.1. Prove that 0J.l- : &int (H) -+ [0, oo) is still a {non-internal) measure, but now with values in 1R rather than *IR . 0 Definition 9.5.2. The Loeb measure LJ.l- is a natural a-additive extension of 0Jl- obtained as follows. If D � H is internal, 0J.1-(D) < oo, X � D, and
sup{ 0J.1-(I) : I � X, I internal} = inf{ 0J.1-(I) : X � I � D, I internal} then X is called Jl--approximable. 1 4 A set X � H is Loeb measurable, or LJ.l--measurable, iff approximable for any D E &int (H) with 0J.l-(D) < oo.
X n D is J.l-
1 4 Take notice that the sup and inf in the displayed formula do exist. Indeed it
follows from Theorem 1.1.9 that every set Y � IR belongs to WIF, thus one may execute the operations in WIF, a ZFC universe.
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.C(J.L) is the family of all LJ.L-measurable sets. LJ.L(X) = sup { 0J.L{I) : I � X, I internal} E [0, +oo) is the Loeb measure of any X E .C{J.L) . The Loeb measure LJ.L is finite iff LJ.L{H) < oo - and this 0 the case is if and only if the original J.L{H) is a limited *-real.
Theorem 9.5.3. Under the above assumptions .C(J.L) is a a-algebra contain ing all Borel subsets of H, LJ.L is a a-additive measure on .C(J.L) satisfying LJ.L{X) = 0J.L(X) for all X E .9int {H), and a set X � H is LJ.L-measur able iff X n D is LJ.L-measurable (or, equivalently, LJ.L-approximable) for any D E .9int {H) with 0J.L(D) < 00 . Proof (sketch) . There are some pitfalls in the argument, mostly related to the case of infinite measures, regarding those we refer the reader to manuals on Loeb measures like [StrB 86, Cut 94, Ross 97, Gol 98, Loeb 00). Yet the core of the problem is to prove that a countable union of null sets is a null set. Thus suppose that sets Zn � H (n E N) satisfy LJ.L { Zn ) = 0 for all n E N. We have to prove that Z = U ne N Zn also satisfies LJ.L{Z) = 0. In other words, for any real £ > 0, £ E IR, we have to find an internal set X with Z � X � H and J.L(X) � £ . By definition for any n E IN there exists an internal set Xn � H such that Zn � Xn and J.LXn < *£ 2 - n- l . Note that N is a set of standard size (like any other set in WIF). By the Standard Size C h oi ce axiom of HST, the family of sets Xn can be chosen as a whole. Now it follows from Extension (see 9.1.22°) that there exists an internal function � defined on *N and satisfying �(n) = Xn for all n E N. Since � is internal, there exists an infinitely large number N E *N ' IN such that �{n) � H and J.L(�(n)) < *£ 2 - n- l for all n < N. It remains to define X = U n < N �(n) . 0 The collection .C(J.L) of all Loeb-measurable sets Z � H is not a set in HST ; it is too big to be a set. (Indeed take any infinite internal X � Z with infinitesimal counting measure and hence Loeb measure 0. Then any Y � X is Loeb measurable as well, but .9(Y) is not a set in HST by Theorem 1.3.9.) For most applications this does not really matter because by 9.5.4 Loeb measurable sets are sufficiently approximable by Borel sets, and on the other hand Borel [H) is a set by Theorem 9.3.9. Another plausible solution is to argue in a universe of the form ll.. [ D") for a sufficiently large cardinal "' ' in the frameworks of the scheme "WIF � D" [ in ll.. [ D") ] " of § 6.4h: .C(J.L) n ll.. [ D") is a set because the Power Set axiom holds in ll.. [ D") .
Exercise 9.5.4. Prove that in the assumptions above if J.L(H) is limited then for any Loeb measurable set X � H there exist a E? set A and a II? set o B such that A � X � B and LJ.L(A) = LJ.L(B) {and hence = LJ.L(X) ) . Example 9.5.5. { 1 ) Suppose that H = { 1, 2, 3, ... , h} , where h E *N ' N. Put J.L(X) = iif- (where #X E *IN is the number of elements of X in D ). Then LJ.L is called the uniform probability Loeb measure on H .
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{2) Let H be as in {1) and J.L(X) = #X for any internal X � H. Then all sets X � H are LJ.L-measurable, with LJ.L{X) = #X for X finite, and LJ.L(X) = oo otherwise. In particular LJ.L( { x } ) = 1 for any singleton. {3) Let K = { 1, 2, 3, ... , k} , where k E *N ' N is a fixed number, and J.L(X) = #kX for any X � H = K x K. Then LJ.L{H) = oo , of course. We observe that 9.5.4 fails in this case. Indeed let Y � K be any non-Borel El set (we refer to Theorem 9.3.9{v)(2)). Then X = K x Y is still a El set in H, and hence Loeb measurable (see § 9.5b ). Suppose on the contrary that B � H is a Borel set with LJ.L(BL::i X ) = 0. There is internal U � H with Bl::iX � U and LJ.L(U) < 0.5, that is #U < �- As #K = k, there is a number 1 � j � k such that U does not contain a pair of the form ( x, j) , x E K. Then the cross-sections Xj and Bj coincide. However Xj = Y is non-Borel while Bj is Borel together with B. Note that LJ.L is point-vanishing in this case: LJ.L( { x } ) = 0 for any x E H. For a measure LJ.L as in {2), the approximation of the type defined in 9.5.4 is possible only for internal X because for LJ.L(AL::i B ) = 0 it is necessary o that A = B .
9.5b Loeb measurability of projective sets It follows from Theorem 9.5.3 that all Borel sets are Loeb measurable. What about more complicated, say projective, sets ? The question of measurability of projective sets has been substantially studied by Polish descriptive set theory. In particular it is well known that 1 o. All E} and all IIl subsets of Polish spaces are measurable in the sense of any finite Borel measure. (A finite Borel measure on a Polish space !C is a a-additive measure ..\ on !C such that any Borel set B � !C is measurable with ..\(X) < oo, and any U � !C is measurable if and only if there exist Borel sets B, B' � !C such that B � U � B' and ..\(B) = ..\(B ' ) - and then ..\(U) = ..\(B) = ..\(B' ) .) 2°. The measurability of sets in higher projective classes is undecidable in the following sense: (a) it is consistent with the axioms of ZFC that there is a A� set of reals X non-measurable in the sense of the Lebesgue measure of IR ; {b) it is also consistent with ZFC that all projective subsets of Polish spaces are measurable in the sense of any finite Borel measure. Let us employ shadow maps to infer analogous results for Loeb measures. We keep here the notation of § 9.5a.
Exercise 9.5.6. Assume, to be on the safe side, that LJ.L is a finite Loeb measure. Let cp : H -+ 2N be a shadow map in the sense of 9.2.12. Define a measure ..\ on 2N as follows: ..\(U) = LJ.L('l/J -1 (U)) for any U � 2N such
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that LJ..t ('l/J -1 (U)) is defined. Prove that then ..\ is a a-additive Borel measure 0 defined on the a-algebra £{A) = {U : 1/J -1 {U) E .C{J..t) } .
Corollary 9.5.7. All sets in Et and IIl are Loeb measurable. Proof (we consider only the case of finite Loeb measures LJ..t ) . Let X � H El set. In particular X is countably determined, that is X = u {Bn}ne N = cp- 1 {U} , where U � 2N , fA = {Bn }ne N is a system of internal subsets of H, and cp = sh� . We can assume that U = cp "X, and then U is E� together with X by Theorem 9.3.3. Define a Borel a-additive measure A on 2 N as in 9.5.6. Then U, as a E� be a
set, is A-measurable by 1 o , and hence X is LJ..t-measurable by the result of 9.5.6, as required. 0
There is another proof of this result, not related to any facts in Pol ish descriptive set theory. Recall that the class El coincides with the class A · f!ii'int (H ) of all sets obtained by action of the Souslin operation A on in ternal subsets of H {Proposition 9. 1.10). However it is known that, by a very general theorem of measure theory due to Marczewski 15 , under certain conditions (definitely satisfied in this case) the class of all measurable sets is closed w. r. t. the action of A, as required.
Exercise 9.5.8. By analogy with 9.5.7, prove a suitable consequence of 2°
for Loeb measurability of projective sets in internal domains.
0
9.5c Approximations almost everywhere It is a typical question of measure theory whether a given rather complicated object can be "approximated" by a rather simple object so that the domain where the two differ from each other is "small" , for instance, is a null set in the sense of a certain measure. The following theorem is an example of this kind in "hyperfinite" descriptive set theory.
Theorem 9.5.9. Let H, K be infinite internal sets, and P � H x K is a set in El or in rrt such that all cross-sections Px are internal. Suppose also that LJ..t is a Loeb measure on H such that LJ..t (H ) is finite. Then there is an internal set Q � H x K such that LJ..t ( {x E H : Px f. Q :z: } ) = 0 . Proof. We can assume that P is IIl; the result for E l follows by taking complements. It follows from Theorem 9.4. 10{ii) that P = U n pn , where pn = An t En , An � H x K are internal sets, and En = {x : A� � Px} = {x : Vy {(x, y) E An => (x, y) E P)} are IIi subsets of H since P is IIi , and hence LJ..t-measurable by Theo rem 9.5.7. Fix a real e > 0. Then for any n E N there exists an internal
1 5 See, for instance, [KurM 76, 7a in XII.8], [Cohn 80, 8.4. 1], or [Rog 70, Thm. 26].
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set Xn (e) � En with LJL(Rn(e)) < e 2 - n , where Rn(e) = En ' Xn (e). Thus we still have Px = Un Q n (e) z for all x E H(e ) = H ' R(e), where R(e) = Une N Rn(e) and each Q n (e) = An t Xn (e) is an internal set. Thus by Saturation for any x E H(e) there is a number n (x ) E N such that already Px = Un < n(z } Q n (e) z . However, as all Q n (e) are internal, the sets Cm = {x E H(e) : Px = Un < m Q n (e)z } are Borel, and H (e ) = Um e N Cm by the above. It follows that there is an internal set Y(e) � H(e) and a number m(e) E N such that LJL(H(e) ' Y(e)) < e and Y(e) � Cm (e } · Then Q(e) = Un < m (e} (An t (Xn(e) n Y(e))) is an internal subset of P satisfying Px = Q(e) z for all x E Y(e). We observe that LJL(H ' Y(e)) � 2e. Taking e = �, n E N, in this construction, we obtain an increasing sequence of internal sets Yn = Y ( �) � H and an increasing sequence of internal sets Q n = Q ( �) � P such that LJL(H ' Yn ) � �, dom Q n � Ym and Px = Q� for all x E Yn , and hence Q�+ 1 = Q� for all x E Yn . By Extension {see 9. 1.2), the sequences {Q n } ne N and {Yn }ne N admit internal extensions {Q n } ne •N and {Yn } n e ·N · Then there is a number h E *N ' N such that the reduced sequences {Q n } n5h and {Yn}n5h are still increasing sequences of internal subsets of resp. H x K and H such that for any n � h the following holds: LJL(H ' Yn ) � � , dom Q n � Yn , and Q�+ 1 = Q� for all x E Yn · It follows that LJL(H ' Yh) � 0 and we have Yn � Yh, Q: = Q� = Px for all n E N and x E Yn . Thus Q: = Px for all 0 x E Y = Une N Yn , therefore the set Q = Q h is as required.
Corollary 9.5.10. If P : H -+ K is a Ill or El function 1 6 , partial or total, then there is a total internal function F : H -+ K such that the set D = {x E dom P : P(x) f. F(x)} satisfies LJL(D) = 0 . 0 A set Q as in the theorem is called an internal LJL-lijting of P. The notion of E?-lijting {of a set with E? cross-sections) , as well as of r-lifting for any other class r, can be defined similarly.
Exercise 9.5.11. Prove, following the proof of the theorem, that any Ill set with E� cross-sections admits a E? lifting. 0 Problem 9.5.12 ("large sections") . In Polish descriptive set theory, any
Borel set in a product of two Polish spaces, all of whose non-empty cross sections have non-zero measure (different version: are non-meager) is Borel uniformizable and hence its projection is Borel too (see [Kech 95, § 18B]). Are there any reasonable analogies in "hyperfinite" descriptive set theory ? For instance, if P � H x K is a Borel set and LJL(Px ) > 0 for every non empty cross-section Px , where LJL is a fixed Loeb measure on K, does it follow that P is Borel-uniformizable (and then dom P is Borel, too) ? This question may be more interesting and difficult in the case of category since in general there exist difficulties with the Baire category notions in 0 "hyperfinite" descriptive set theory, see [KL 91). 16
We identify any function with its graph.
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9.5d Randomness in a hyperfinite domain It sometimes happens in mathematics that an intuitive notion cannot be easily formalized so that both the spirit and the letter is kept. The notion of a random object (for instance a random real) is among those notions. The approach determined by classical probability theory simply dismisses as nonsense the concept of a single random real. Different attempts were made to introduce an adequate definition of a single random real, mostly in the frameworks of recursion theory. Their com mon denominator is as follows: a real x is defined to be random if no infinite amount of information about x is available. A similar notion is known in set theory: a real is called random, or Solovay-random, over a given model 9J1 of ZFC if it avoids any Borel set of measure zero coded in !D1. We attempt here to give a reasonable notion of randomness in HST. Our approach has some semblance of the Solovay randomness, but we employ the standard universe S, and universes of the form S[w] , w E D, in the role of a ground model in the Solovay randomness. Recall that S[w) consists of sets of the form f(w) , where f is a standard function such that w E dom f, and sets in S[w) are called w-standard.
Lemma 9.5.13. Let Y, w be internal sets. Then the set Y' = Y n S[w) of all w-standard elements y E Y is a set of standard size. Proof. There exist standard sets W and S such that Y � S and w E W. Let F be the standard set of all internal functions f : W -+ S. Each y E Y' has the form f (w) for some f E F n S, which is a set of standard size. 0 Definition 9.5.14. Let w be an internal set. Say that a *-real x is w-infinitely large iff x ;:::: c for some w-standard infinitely large c > 0; w-infinitesimal iff lxl � £ for some w-standard infinitesimal £. 1 7 Suppose that H is a hyperfinite internal set and J1- .9int (H) -+ *[0, 1) an :
internal *-finitely additive probability measure on H. An element x E H is w-random w. r. t. J1- iff x does not belong to any (H, w)-standard set X � H with H -infinitesimal value J.l-(X). 0 (The postfix "w. r. t. Jl-" can be omitted if this does not lead to ambiguity.) The following lemma shows that, in agreement with intuition, non-random elements form a scattered family.
Lemma 9.5.15. In the assumptions of 9.5. 14 the collection !JP,C of all x E H non-w-random w. r. t. J1- can be covered by an internal set X � H with infinitesimal J.l-(X), and hence LJ.l-(!JP,C ) = 0 . 1 7 The definition of w-infinitely large and w-infinitesimal reals makes sense iff there really exist w-standard infinitesimals and infinitely large numbers. In particular it does not make sense (and will not be used) in the case when w is standard.
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It is not asserted here that J.L(X) is e. g. H-infinitesimal. We cannot claim that J.L(!Jf.C ) itself is infinitesimal because J.L is defined only for internal subsets of H while !Jf. is, generally speaking, external.
Proof. It follows from Lemma 9.5.13 that the collection J of all (H, w) standard sets I � H, such that J.L(I) is H-infinitesimal, is a standard size subset of the internal power set P = &'int (H) (which is a hyperfinite set). By Saturation , there is an infinitesimal £ bigger than all numbers J.L( I) , where I E J. Also by Saturation, there is an internal set / � P containing � 1/ .ji elements and satisfying J � /. We can assume that J.L( J) < £ for all J E / (otherwise / can be accordingly restricted). Then the internal set X = U / satisfies J.L(X) < .ji. On the other hand !Jf.C � X . 0 In principle it can be required, in the lemma, that in addition J.L(X) < 8, where 8 is an arbitrary but fixed *-real (perhaps, infinitesimal) bigger than all h-infinitesimals.
Example 9.5.16. (1} Let H = int2h (the internal set of all internal maps s : h -+ 2). Obviously #H = 2h in D, so we can define the counting measure TJ on H by TJ(X) = #(X) 2-h for all internal X � H. Sequences s E H random w. r. t. TJ can be called uniformly random. (2) Consider the set fA = {0, 1, ... , h} , with the Bernoulli measure {3, defined on singletons by {3( { k}) = 2-h (Z) . Numbers k E {0, ... , h} random w. r. t. {3 can be called Bernoulli random. 0 Exercise 9.5. 17. Let s E H be uniformly w-random. Prove that k(s) = #{ n : s(n) = 1 } E fA is a Bernoulli w-random number. 0 Theorem 9.5.18 (Fubini) . Suppose that w is an internal set, H, K are infinite *-finite sets, while J.L, v are internal finitely additive probability mea sures on resp. H, K, and v is (w, H, K)-standard. Then (i) if (x, y) E H x K is w-random in H x K w. r. t. J.L x v then x is w-random in H w. r. t. J.L while y is (w, x)-random in K w. r. t. v ; {ii) if K is H-standard, H is K-standard, x E H is w-random in H w. r. t. J.L, and y E K is (w, x)-random in K w. r. t. v, then (x, y) is w-random in H x K w. r. t. J.L x v .
Proof. (i) Let X � H be a (w, H)-standard set of measure J.L(X) < £ , where £ is H-infinitesimal. Assume on the contrary that x E X. Then (x, y) E P, where P = X x K is (w, H, K)-standard and satisfies (J.L x v) (P) < £, which is a contradiction. (Note, in passing by, that to be (w, H, K)-standard and to be (w, H x K)-standard is one and the same.) Let Y � K be a (w, x, K)-standard set of measure v(Y) < £, where £ is K-infinitesimal. Suppose on the contrary that y E Y. By definition we have Y = f(w, x, K) , where f is a standard function. Let P be the set of all pairs
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(x', y' ) E H x K such that Yx' = f (w , x', K) is a subset of K satisfying the inequality v(Yx') < £, and y' E Yx' · Note that P is (w, H, K)-standard by the assumptions above, and (J.L x v) (P) � £. On the other hand, (x, y ) E P by definition, which is a contradiction. (ii) Consider a (w, H, K)-standard set P � H x K with (J.L x v) (P ) < £ , where £ is a (H, K)-infinitesimal; hence H-infinitesimal because K is H standard. Put Px' = { y E K : (x' , y) E P } for any x' E H. Under our assumptions the set X = { x' E H v(Px' ) ;:::: J€} is (w, H)-standard, and J.L(X) � J€ because (J.L x v) (P) < £. (A "discrete" version of Fubini theorem is applied.) Therefore x ¢ X by the randomness of x. Thus the (w, H, x) standard {therefore (w, K, x)-standard) set Y = Px satisfies v(Y) < J€. However y E Y, which contradicts the randomness of y . 0 :
Corollary 9.5.19 (Steinitz Exchange) . Under the assumptions of Theo rem 9.5. 18 suppose that K is H -standard, H is K-standard, and both J.L and v are (w, H, K ) -standard. Then, if x E H is w-random w. r. t. J.L and y E K is (w, x) -random w. r. t. v then x is (w, y ) -random w. r. t. J.L . 0 9.5e Law of Large Numbers In classical probability theory, this is a common name for several important theorems saying that under some conditions the arithmetic mean Et + ·�+s" of jointly independent random variables €i is close to the arithmetic mean of their expectations m1 + ·�+mn . (See [Sin 93), Section 12.) Our goal here is to obtain a "hyperfinite" version, based on the notion of randomnes introduced in § 9.5d. Suppose that J.L an internal finitely additive probability measure on a *-finite set H, as above. Assume, in addition, that H � *IR. We define = L: xeH x J.L( { x} ) , the expectation of J,L; 2 Var J.L = L: x eH (x - E J.L) J.L({x}}, the variance of J.L.
E J.L
Note that the expectation and variance are functions of the measure ( = the probability distribution) rather than of random elements as we defined them. Suppose that h E *N ' N, and for any n = 1, 2, . . . h, Hn � *IR is a *-finite set and J.Ln is an internal finitely additive probability measure on Hn, such that the maps n � Hn and n � J.Ln are internal. We put mn = E J.L n and Vn = Var J.L n for all n. Define H = rr := l Hn {the product consists of all internal functions f defined on { 1, 2, . . . h} so that f (n) E Hn, \1 n ) and let J.L = rr := l J.Ln be the internal product probability measure on H .
Theorem 9.5.20 (Hyperfinite Law of Large Numbers) . Under these assump tions, if v = h- 1 L:: = l Vn is a limited number and the measure J.L is H standard then, for any sequence x = { xn }�= l , random (i. e. 0-random) in H w. r. t. J.L, the following difference is infinitesimal:
9.5 Loeb measures
x + Ll(x) = 1
·
·
·
h
+ xh
m + - 1
·
·
·
+ mh
----
h
359
.
Proof. By Kolmogorov 's inequality (see e. g. [Sin 93], Theorem 12.2), applied in the internal universe, we have v J.L { { y E H Ll (y) � s}} � 2 hs for any s > 0. By the assumption, vs - 2 is a limited number whenever s > 0 is standard. Thus the set X8 = { y E H : Ll(y) � s} has an h-infinitesimal measure J.L (X8 ) whenever s > 0 is standard. On the other hand, if s is standard then X8 is (H, J.L)-standard, and hence H-standard because J.L is assumed to be H-standard. We conclude, by definition, that x Ff. X8 for any standard s > 0, as required. 0 :
•
9.5f Random sequences and hyperfinite gambling There exists another idea of randomness. One may view a binary infinite sequence a E 2w as random if a human cannot win an unlimited amount of money in gambling against a. In HST, this idea can be realized by a certain game of *-finite length. Fix a number h E *N ' N. The set .!7 = int { -1, 1 } h of all internal sequences of the form a = (a0, a2 , ... , ah - 1 ) , where each ai is -1 or 1 , is an internal *-finite set of *-cardinality #.!7 = 2h in D. Every set A � .!7, not necessarily internal, defines a game G (A) between two players, the Gambler and the Cas ino, which proceeds in D, the internal universe, as follows. Gambler has, at the beginning, an initial amount of money, Bo = $ 1 . A run in this game consists of h steps. At each step n = 0, 1, 2, ... , h - 1 : bets an amount of money bn , a *-real satisfying l bnl � Bn , as to the result of Casino's forthcoming move an E { -1, 1} ; 2) Casino observes bn and moves an = -1 or 1 ; 3) Gambler's next balance Bn+ l is computed by Bn+ l = Bn + bn an· {In other words, say bn = -0.75 means that Gambler bets $ 0. 75 on the move an = -1. If Casino in fact plays an = -1 then Gambler wins $ 0. 75 at this step, otherwise Gambler loses this amount.)
1)
Gambler
This results in an internal sequence a = ah = (ao , ... , ah- 1 ) E .!7 of Casino's moves, and the final Gambler's balance Bh , a nonnegative *-real. The Casino's goal in this game is to produce a sequence a E .!7 which belongs to A; the Gambler's aim is, by betting money, either to force Cas ino to produce a Ff. A or to gain a large enough amount of money if Casino is willing to reach A by all means. Who wins the game in the case a E A depends on a definition of what is the "large enough" final balance Bh to determine Gambler 's win. See Definition 9.5.22 below on possible version.
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9 "Hyperfinite, descriptive set theory
Thus the role of the set A in the game is the following: Casino must play so that ah E A in order not to lose independently of the balance score. Therefore Gambler can exploit the unability of Cas ino to play in an absolutely free way, and make reasonable predictions aiming at increasing the balance. It is intuitively clear that the larger A is the easier Casino's task should be, and the other way around for Gambler. In quantitative terms, this is expressed by the following result of [KanR 96a) given here without a full proof. Put TJ(X) = #X 2 - h for any internal set X � Y; this is a counting measure on Y. Let LTJ denote the corresponding Loeb measure. Theorem 9.5.21. {i) A � Y is a set of Loeb measure LTJ (A) = 0 if and
only if Gambler has an internal strategy 18 in G(A) which guarantees that Bh is infinitely large whichever way Cas ino plays. (ii) Let r be a positive *-real. Then A has an internal superset of counting measure r - 1 if and only if Gambler has an internal strategy in G(A) which guarantees Bh ;:::: r in G(A) . (iii) Let r be a positive *-real. Then A has an internal subset of counting measure r - 1 if and only if Casino has an internal strategy in G(A) 0 which guarantees Bh � r in G(A) .
Proof (sketch) . {i) Assume, for the sake of simplicity, that A is internal. Let £ = TJ (A) . For any t E int { -1, 1}" ( n � h ) put Yt = {a E Y : t C a} and dt = TJ(� n A) 2 - n , the density of A on Yt. Thus for instance dA = £. ( A is the empty sequence.) Suppose that n < h and t = (a0 , ... , an_ 1 ) is the sequence of Casino's n initial moves. Obviously dt �� - 1 = dt - r and dt 11 1 = dt + r for some (positive or negative) *-real . r = Tt , l rt l � dt . Gambler's optimal strategy is to bet bn = Bn !:L dr , so that dBn tl d independently of an = !la. the ensuing Casino's move an · Playing this way, Gambler has lJ!" = .!]: = £- 1 in the end of the run. That is, if Casino has played a E A (and otherwise Casino has lost) then the final density do: is equal to 1, and hence Bh , the final balance, is equal to £- 1 • In other words Gambler has a strategy that guarantees Bh ;:::: £- 1 . This proves ==> in {i). To prove b, where b is an infinitely large h-standard *-integer. Thus a belongs to the (h, w) standard set A = {a' E .!7 : B(a' ) > b} � .!7. Finally, TJ(A) < b- 1 because L: a' e.9' B {a' ) = 2h . It follows that TJ(A ) is h-infinitesimal, as required. 0 Recall that a game is determined if one of the players has a winning strategy. For instance for any *-real r ;:::: 0, if A � .!7 is a LrJ-measurable set then the game G(A)r , that is G(A) specified so that Gambler wins whenever aH ¢ A or Bh ;:::: r, is determined by Theorem 9.5.21. In Polish descriptive set theory any Borel game of length N is determined (see e. g. [Kech 95, 20.C]). 'Borel ' here means that the set A � N N , which defines the result in the sense that player I wins iff the final sequence a = { an }ne N of moves belongs to A. But in "hyperfinite" descriptive set theory such a Borel determinacy badly fails. Exercise 9.5.24. Fix h E *N ' N. Any set A � H int22 h {internal func tions { 0, 1, ... , 2h - 1 } -+ 2 ) defines a game GA in which player I makes moves a0 , a2 , . .. , a2 h- 2 E { 0, 1}, player II makes moves a 1 , a3 , ... , a2 h- 1 E { 0, 1}, and I wins iff the sequence {ak}k here with Theorem 1.4.9. 2 1 The result is a corollary of Theorem 1 . 4.6 ( i ) , of course, but we are interested to give a proof using only countable Satu ration.
9.6 Borel and countably determined cardinalities
365
which includes X. By (countable) Saturation, u = cut u = Ut e F n m Ut tm, where Us cut Xs , hence, Us = [0, J.Ls], where J.Ls = m ax Xs E *N for all < s E 2 w . If there is f E F with U n m Ut tm then the sequence {ht tm}me N witnesses that U is countably coinitial, or contains a maximal element if the sequence is eventually constant. Otherwise, by Saturation, for any f E F there is m1 E N such that ht tm1 E U. Let S = {! t mt : f E F}; this is a countable set and easily U = nse s[O, J.Ls], so that U is countably cofinal. o =
=
Lemma 9.6.8. Suppose that An , Bn are *-finite internal sets, and bn = #Bn � an = #An for each n. Then {i) if A n+ l � An and Bn+ l � Bn for each n then nn Bn �s nn An ; {ii) if An � An+l and Bn � Bn+ l for each n then Un Bn � B Un A n . Thus � B is sometimes preserved under unions and intersections ! Proof. {i) For any n there is an internal bijection f : A0 ontS [0, a0) such that f " Ak = [0, ak) for all k � n. By Saturation, there is an internal bijection f : Ao ont� [0, ao) with f" An = [0, an) for all n E N. We conclude that nn An =s A = n n an. Also, nn Bn =s B = nn bn. However B � A. (ii) Arguing the same way, we prove that Un An =s A and Un Bn =s B, where B Un bn � A = Un an . 0 =
If U � V � *N are cuts then we write U � V iff � � 1 for all x, y E v, u. (For instance if U = [0, a) and V = [0, b) then U � V iff � � 1. ) This turns out to be a necessary and sufficient condition for U :::: 8 V.
Lemma 9.6.9. (i) H U, V are Borel cuts then U :::: 8 V iff U � V. (ii) Any �-class of Borel cuts contains a �-minimal cut. (iii) Any additive Borel cut is �-isolated, i. e., U ¢ V for any cut V i:. U . Proof. (i) Let, say, U � V. Suppose that U =s V. Take any x < y i n V ' U. Then x :::: 8 y, hence, �y � 1 by Theorem 9.6.4. To prove the converse suppose that U � V. Take any c E V ' U. Let A = {a E *N : % � 0}. We observe that A � U (indeed, the entire part of � belongs to U ' A). Put x+ = {c + a : a E A} and x- = {c - a : a E A } . Obviously U' = [O, c) ' x- � U and V � V' = [0, c) U x+ , and hence it suffices to define a Borel bijection U' ont� V'. Let Z = [O, c) , A. Then U' = Z U A and V' = Z U A ux + ux- , where the unions are pairwise disjoint. Exercise: prove that the map
F(z) =
{
z, x, c - x, c+x,
whenever z E Z whenever z = 3x E A whenever z = 3x + 1 E A whenever z = 3x + 2 E A
366
9 "Hyperfinite" descriptive set theory
is a Borel bij__ection U' ontS V'. (ii) Let U be the set of all x E U such that there is y E U, y > x with � 'fi 1. This is a cut, moreover, a projective set, hence, countably determined, which implies that fJ is actuallY Borel by LemmaJ.6. 7. Moreover � � U. Finally, note that for any x E U there exists x' E U, x' > x, with � '/:. 1 : indeed, let x' = :z:�u , where y E U, y > x, � '/:. 1. This suffices to infer that V ¢ fJ for any cut V � fJ. In other words, fJ is the �-least cut =s-equivalent to U, as required. 0 (iii} That fJ = U for any additive cut U is a simple exercise.
9.6c Proof of the theorem on Borel cardinalities Here we prove Theorem 9.6.6. Lemma 9.6.9 allows us to concentrate on the first assertion of the theorem. Since all Borel sets are countably determined, we can present a given Borel set X � *N in the form X = UleF nn Xnn , where F and the internal sets Xs � *N are as in {2) of § 9.2b. In accordance with 9.6.3, we can assume that X is bounded in *N then it can be assumed that all sets X8 are also bounded, and hence *-finite. Let v8 = #Xs . Let C be the set of all e E *N such that there is f E F and an internal injection cp : [0, e} -+ X1 = nn Xf tn. Then C is a cut and a countably de termined set. (By Saturation, for any internal Y to be internally embeddable in X1 it suffices that #Y � Vftm for any m. ) We claim that C �B X. Indeed if there is f E F such that C � [0, vl rn) for all n then immediately C �8 X1 by Lemma 9.6.8{i}. Otherwise for any f E F there is n1 E N such that v1 tn 1 E C. As X1 tn 1 is an internal set with #Xnn 1 = vnn1 , no internal set Y with #Y > vnn 1 admits an internal injection in X1 . Thus the countable set { v1 tn1 : f E F} is cofinal in C, and hence C = U k Zk , where all Zk belong to C. However for any k there is an internal Rk � X with #Rk = Zk· Lemma 9.6.8{ii) implies c �B u k Rk . In continuation of the proof of the theorem, we have the following cases. Case 1 : C is not additive. Then there is e E C such that eN = U and 2e ¢ C. Prove that X �8 eN. By Lemma 9.6.8{ii), it suffices to cover X by a countable union U3 lj of internal sets l'j with #lj � 2e for all j. For this it suffices to prove that for any f E F there is m such that Vftm = #Xnm � 2e. To prove this, assume, on the contrary, that f E F and vl tm 2= 2e for all m; we obtain, by Saturation, an internal subset Y � X1 with #Y = 2e ¢ C, contradiction. We return to this case below. In the remainder, we assume that C is additive. Case 2: C is countably cofinal. Arguing as in Case 1, we find that for any f E F there is m such that vl tm = #XI tm E C. (Otherwise, using Saturation and the assumption of countable cofinality, we obtain an internal subset Y � X1 with #Y ¢ C, contradiction.) Thus, X can be covered by a -
9.6 Borel and countably determined cardinalities
367
countable union U Yj of internal sets Yj with #Yi E C for all j. It follows, by Lemma 9.6.8{iiJ,i that X �8 C. Since C �8 X has been established, we have X :::: 8 C, so that U C proves the theorem. Case 3: C is countably coinitial, and there exists a decreasing sequence h {hk } k et-·h coinitial in *N , U, such that ....! h !.L is infinitesimal for all k E N. For any k E N , if f E F then there is m with Vf tm � hk + l (otherwise, by Saturation, Xt contains an internal subset Y with #Y > hk+ l ' contradiction), so that X is covered by a countable union of internal sets Yj with #lj � hk+ l h is infinitesimal, that, for all j. It follows, by Saturation and because _!.!Ja_ h for any k, X can be covered by an internal set Rk with #Rk � hk . Now X �B C by Lemma 9.6.8(i), hence, U = C proves the theorem. Case 4: finally, C = c/N for some c f/. U. We have c/N �8 X �8 eN (similarly to Case 2). We finish the proof. Cases 2 and 3 led us directly to the result required, while cases 1 and 4 can be summarized as follows: there is a number c E � 'N such that c/N �8 X �8 eN . We can assume that X � eN. Let 7Jn(Y) = if[- be the counting measure on the interval [nc, nc + c) for any n E N, and L7]n the corresponding Loeb measure. For any Z � eN such that Zn = Z n [nc, nc + c) is L7]n-measurable for all n E N, put L77(Z) = L: ne N L7Jn(Zn) · The set X is Borel, thus L77(X) is defined. H L77(X) = oo then there is a sequence { Xn} of internal subsets of X with #Xn = nc, V n. It follows that eN �8 X by Lemma 9.6.8, hence, X =s U = eN, as required. Suppose that L77(X) = r < oo. There is an increasing sequence {An}ne N of internal subsets of X and a decreasing sequence { Bn}ne N of supersets of X such that 7J(Bn) - 1](An) -+ 0 as n -+ oo (i. e., the difference is eventually less than any fixed real c > 0). If r = 0 then #�n -+ 0, thus nn Bn =s c/N by Lemma 9.6.8, which implies X :::: 8 c/N since c/N �8 X, therefore, U = c/N proves the theorem. Finally, assume that r > 0. Prove that then X :::: 8 E(cr) {the entire part of cr). We have #�n -+ r from below and #�n -+ r from above. Let u = U ne N #An and v = n ne N #Bn; then U n An =s u and n n Bn =s v by Lemma 9.6.8, while E(cr) E V ' U, hence, in remains to prove that U :::: 8 V. By Lemma 9.6.9, it suffices to show that U � V. Let x < y belong then 11. � is not infinitesimal, which contradicts the to V ' U. H 1l. � r- 1 fact that # Bn 1tb.. -+ 0 because 1tb.. < � and 1L < # Bn for all n . =
lo - 1
11 - 1
:z:
c
-
c
c
-
c
c
-
c
c
-
c
0 (Theorem 9. 6. 6)
Corollary 9.6.10. (i) Any two Borel sets X, Y � *N are �8-comparable. (ii) H c E *N ' N and X, Y � eN are Borel sets of non-0 measure L77 {see the proof of the theorem) then X :::: 8 Y iff L77(X) = L77(Y) . Proof. (ii) See the last paragraph of the proof of the theorem.
0
368
9 "Hyperfinite" descriptive set theory
9.6d Complete classification of Borel cardinalities Call a Borel cut U � *N minimal if V �8 U for any cut V � U. It follows from Theorem 9.6.6 that any =:8-class of Borel subsets of *N contains a unique minimal Borel cut, so that minimal Borel cuts can be viewed as B orel cardinals (of Borel subsets of *N ) . For instance, any additive Borel cut is minimal by Lemma 9.6.9, hence such a cut is a Borel cardinal. But if U is a non-additive minimal Borel cut, then there is a number c E U with 2c f/. U, so that c/N � U C eN, and, accordingly, c/N x E y ; (II) there is an internal pairwise E-inequivalent set Y � H with #Y ¢ U . Moreover, if (II) holds and U satisfies x E U => 2:z: E U then {I) fails even for countably determined maps 19 .
The theorem yields a true dichotomy only for "exponential" cuts U, i. e., those satisfying x E U => 2:z: E U. If this condition fails then (I) and {II) are compatible, for take E to be the equality on [0, 2:z: ) but y E z for all y, z ;:::: 2:z: . It is an open problem to obtain a true dichotomy in the general case. Note that the case U = N in this theorem is equivalent to Theorem 9.7.1. Indeed, for the less trivial direction, a shadow map cp as in 9.7. 1{1) can be transformed to an internal map 19 : H -+ int2c such that cp{x) = 19 {x ) t N for any x E H, see the proof of Theorem 9.7.7. Then 19 {x ) = 19 (y ) implies x E y for all x, y E H, as required.
Exercise 9. 7.9 (difficult !). Prove Theorem 9. 7.8 following the proof of The 0 orem 9.7. 1 with appropriate corrections. There is a somewhat different approach to maps 19 as in (I), which may lead to new insights. Put cp(x) = 19 {x) t U. Then cp is a map defined on H, with values in 2 u , and cp{x) = cp(y) implies x E y. The values of cp are not just arbitrary external maps U -+ 2. Say that a function � : U -+ 2 is internally extendable, in symbols � E {2 u hex, if there exist an internal set Z with U � Z and a map f E int2 Z {that is f : Z -+ 2 is an internal function) such that � = f f U. {If U itself is internal then this is the same as an internal function.) This definition obviously does not depend on the choice of an internal set Z 2 U, that is, we can take Z = [0, c) , c e *IR ' U.
376
9 "Hyperfinite" descriptive set theory
In these terms, we have x E y cp(x) R cp (y ), where � R TJ iff � = TJ, or there exist x, y E H with xEy, is an equivalence relation on {2u hex· However it is difficult to study the relation R and its connection with E by means of "hyperfinite" descriptive set theory simply because the domain {2 u hex of R consists of non-internal objects. Yet we can define the lifting F Rt c of R to c = [0, c) , that is an equivalence relation f F g iff (f t U) R (g t U) on the internal set int2c . Clearly F is countably determined. Then a similar equivalence x E y ¢:::::} 19 (x) F 19(y) holds. This means that 19 is a reduction of E to R (see § 9. 7f on a general definition). Moreover F is concentrated on U in the sense that whether f F g ( f, g E int2c ) depends only on f f U, g f U. We conclude that (I) of Theorem 9.7.8 can be reformulated as follows: (I') there exist c E *N ' U, a countably determined equivalence relation F on int2 c concentrated on U, and an internal reduction 19 of E to F . Note that (I') does not depend on the choice of c, that is if it holds for some c ¢ U then it also holds for any other c' ¢ U. =
9. 7d Transversals of "countable" equivalence relations An equivalence relation E is called "countable" if all of its equivalence classes [x] e = {y : x E y} , x E dom E, are at most countable. In Polish descriptive set theory, "countable" Borel ERs form a rather rich category whose full structure in terms of Borel reducibility is a topic of deep investigations (see [JackKL 02]). In nonstandard setting, the structure of "countable" ERs is much more elementary due to the next theorem. This is another side of the same phenomenon making "planar" sets with countable cross-sections look simpler in "hyperfinite" descriptive set theory than in Polish spaces, see § 9.4b. Recall that a transversal for an equivalence relation is any set having exactly one element in common with each equivalence class.
Theorem 9. 7 .10. Any "countable" countably determined equivalence rela tion E on an internal set H admits a countably determined transversal. Proof. Note that E, as a subset of H x H, is a countably determined set with all cross-sections Ex = [x] e = {y : x E y} being at most countable sets. It follows from Theorem 9.4.7(ii) that there exists a countable sequence { Fk } ke N of internal functions Fk : H -+ H such that E � U k Fk , or in different terms [x] e � { Fk (x) : n E N } for all x E H. The sets Dk = dom ( E n Fk ) = { x E H : x E Fk (x) } are count ably determined (as internal preimages of E, a count ably determined set) . Let us fix any internal well-ordering -< of H. In other words, it is true in 0 that -< is a well-ordering of H. Let n E N. For any x E H we carry out the following construction, called the n-construction for x . Define an internal - X E Smn) A V m 3 n (x E Dn A X E Smn) , where Smn = {x : 1/J{Fn (x)) = Ym}· It is clear that every set Smn is Borel over !C (even over the sets Xmn )· It follows that X(A) is Borel, and hence countably determined, over !C, as required. 0
378
9 "Hyperfinite" descriptive set theory
On the other hand, the class of all sets countably determined over a fixed countable algebra !Z" of sets is closed under any unions (as well as under complements and intersections): to show this take the set theoretic union of the associated bases under the assumption that the assignment of sets in !Z" to indices i E N is fixed once and for all. (Note that the class of all countably determined sets is closed only under countable unions and intersections !) It follows that A(X) is countably determined over !Z" . But !Z" itself consists of countably determined sets, therefore it remains to cite Theorem 9.2.9. 0 {Theorem 9. 7. 10)
Example 9.7.13. The equivalence relation x M N y iff lx - Yl E N on *N is "countable" , and hence it has a countably determined transversal by Theo rem 9.7.10. Note that M N is a rry relation. Thus it is natural to ask whether M N has a transversal of a type simpler than CD. It clearly does not ad mit a Borel transversal - by the same "shift" argument as in the proof of the fact that the Vitali equivalence on IR does not admit a Borel {generally, Lebesgue measurable) transversal. Theorem 9. 7.18 below contains an even stronger result. Whether M N has a projective transversal depends on the Loeb measura bility of projective sets in segments [0, a) of *N (and hence on the Lebesgue measurability in 2N , see § 9.5b) for the negative direction, or projective choice of an element in an arbitrary countable subset of 2N for the positive direction 0 - and hence is independent of HST. 9.7e Equivalence relations of monad partitions This class of equivalence relations was defined in § 1.4c: any additive cut U � *N induces an equivalence relation x Mu y iff lx - Yl E U on *N, which divides *N into Mu-equivalence classes [x]u = {y : x Mu y} = {y : lx - Yl E U}, called U-monads. It follows from Lemma 9.6. 7 that the case of additive countably determined cuts U � *N splits into two subcases: countably cofinal and countably coinitial cuts. (Let alone 0 and *N, the only internal additive cuts.) Accordingly, this leads to the following classes of ERs: countably co.final equivalence relations: those of the form Mu, where U � *N is a countably cofinal additive cut - all of them belong to EY ; countably coinitial equivalence relations: those of the form Mu , where U � *N is a countably coinitial additive cut - all of them belong to rry . We proved {Theorem 1.4. 7) that among these ERs, only those of the form MhN , h E *N, in particular, M N , and those of the form M h/N , h E *N ' N ad mit transversals in the class A28 • As the latter is strictly bigger than the class of all countably determined sets (see 9.2.8), it is a natural question whether the ERs of the form hN and h/N have countably determined transversals. The answer turns out to be different for the two subfamilies.
9. 7 Equivalence relations and quotients
379
Theorem 9.7.14. (i) If 1 � h E *N then MhrN admits a CD transversal; {ii) if h E *N ' N then M h/N does not have a CD transversal. Proof. (i) Note that M N admits a countably determined transversal T, see 9.7.13, or alternatively, the transversal defined in the proof of Theo rem 1.4.7(i) is CD. It follows that any MhN has a countably determined transversal: just take { hx : x E T} . (ii) Let us prove that even the restricted relation Eh = M h/N t [0, h) does not admit a CD transversal. In the notation of the proof of (ii) of Theorem 1.4. 7 (the "if" part), it suffices to show that the equivalence � R TJ iff � t N = TJ t N on int2c (where c E *N ' N ) does not have a countably determined transversal. We claim that moreover any countably determined set X � int2 c inter secting every R-class in a countable set is countable. (Note that 2N is uncount able, and hence so is the set of all R-equivalence classes.) As usual, we have X = U Je F nme N Xn m � int2c , where F � 2 N and all sets Xs , s E 2<w , are internal subsets of int2c . Thus, for any f E F and g E 2N , the intersection of X1 = nme N Xn m (a subset of X ) and Y9 = {� E int2c : � t N = g} (an equivalence class of R) is at most countable. It follows that X1nY9 , a II� set, is in fact finite by 9. 1.7. Therefore, by Lemma 9. 1.5{iii), for any pair of f E F and g E 2N there is a number m E N such that D J rm , 9 rm = XJrm n Y9 rm is a finite set. On the other hand, we have X � Us ,te 2 <w , lh s=lh t D st , so that 0 X is countable, as required. Here follow several corollaries and related results.
Exercise 9. 7 .15. Let h E *N , N and Eh = M h/N t[O, h). Prove the following: {1) For all x, y < h, the equivalence holds: x M h/N y iff zh.y � 0, and hence Eh has exactly c-many equivalence classes (2) Any internal set X � [0, h) which intersects every Eh-class in a finite set is finite. (Hint. If X is infinite then there is a decreasing chain [0, h) = 10 ;2 11 ;2 12 of subintervals whose length tends to 0, but X n 1k still infinite for any finite k. Now apply Lemma 9. 1.5.) (3) The relation x :::: 8 y on *N does not admit a countably determined transversal. (Hint. x =B y is equivalent to xfy � 1 by Theorem 9.6.4, and hence ::::8 coincides with Eh on [ �, h) .) (4) On the contrary, the relation x = en y on *N admits a countably deter mined transversal. For instance, take, using Theorem 9.6.4{ii), a count ably determined transversal for M N - then {2z : x E T} is a countably determined transversal for = en . 0 •••
The next exercise shows that equivalence relations of the form Mu can be presented differently. Such an alternative form will be useful in some ap plications.
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9 "Hyperfinite11 descriptive set theory
Exercise 9.7. 16. Let S = {� E 2N : {k : �{k) = 1} is finite} , accordingly, *S is the set of all internal functions � E 2 ·N such that { v E *N : � (v ) = 1} is *-finite.; *� is internal. Suppose that C � *N is a non-internal cut. Put, for �, 1J E 1nt2 N , � Rc 1J iff � r (.N ' C) = 1J t (*N ' C) . {1) Prove that 2c = Ucec [O, 2c ) is an additive cut and the map � �---+ s(�) = L v e •N 2"�(v ) is an isomorphism of ( *S ; Rc) onto (*N ; M 2c ) . To restrict this phenomenon to a *-finite domain, let us choose any m E *N satisfying C � [0, m) . {2) Prove that the map � 1---+ s(�) = L v < m 2"�(v) is a bijection of int2 m onto [0, 2m ) satisfying � t { [0, m) ' C) = 1J t { [0, m) ' C) iff s(�) M 2c s(fJ ) . Finally, we put W = { m - x : x E [0, m ) ' C} , the dual cut. Recall that Dw tm is the lifting of the equality Dw on the set {2 w hex of all internally extendable maps a : W -+ 2 to m, that is an equivalence relation on int2m such that � Dw1m TJ iff � t W = 1J f W (see § 9.7c). {3) Prove that � �---+ s'(�) = L v< m 2"�(m - v) sends Dw tm onto M 2 c . Thus we may consider the quotient [0, m)/M 2c as an adequate "model" of 0 {2 w h ex in "hyperfinite" descriptive set theory. 9.7f Borel and countably determined reducibility The following definition is copied from studies in Polish DST. Suppose that E, F are equivalence relations (ERs, for brevity) on Borel sets X, Y. We write E �8 F, in words: E is Borel-reducible to F, iff there is a Borel map (called: reduction ) lx - x' l < k } , m, k E N . Note that Um'k' � Um k whenever m � m' , k � k'. Thus by Saturation in the form of Lemma 9. 1.5{i), we find numbers m o , ko E N such that, for all x , x' E X and y, y' E R : x Cm0 y A x' Cm0 y' A y = y' ==> lx - x' I < ko . Further, changing ¢:::::} to ==> in the subformula in {1), we have
{2)
382
9 "Hyperfinite" descriptive set theory V m (x Cm y A x'
Cm y' ) ==> y = y'
whenever x, x' E X and y, y' E R satisfy l x - x' l < 4k0 • Applying Satur ation as above, we find a number m 1 ;:::: m0 such that x Cm1 y A x' Cm1 y' A l x - x' I < 4ko ==> y = y'
{3)
holds for all x, x' E X and y, y' E R. Now we claim that l x - x' I < k V I x - x' I ;:::: 4k
for all x, x' E X ,
(4)
which obviously contradicts the assumption LJ,L{X) ;:::: !· To prove (4) put y = 19(x) = 19' (x) and y' = 19(x') = 19'{x'), thus x Cm y A x' Cml y' for all m. H y = y ' it then follows from (2) that lx - x'l < ko . H y = y ' then lx - x' l > 4ko by {3). 0
Exercise 9.7. 19. {1) Show that Theorem 9.7.18 implies that there exists no Borel transversal for M N . {2) Prove o.N �B M N , and hence D ·N � f2: an = �� t 2: an ·
{1)
A similar (symmetric) argument also yields the following:
3 k 3 m V �, �� , fJ, TJ1 E *S : {2) � Cn m TJ A �� Cnm fJ1 A � t 2:an = �� t 2: an => TJ t 2: bk = fJ1 f 2:b�o • Suppose, towards the contrary, that rate U � rate V, thus rate V � rate U. Then N � U, and hence U is a fast cut. We can suppose that an+ l - an is infinitely large for all n. As rate V � rate U, there is an index k such that the sequence { bk' - bk } k' > k is not cofinal in rate U. Let n, m be a pair of numbers satisfying {1) for this k. By the choice of k, there exists a number n1 > n such that an' - an > bk' - bk for any k1 > k, hence, in fact, an' - an > f. + bk' - bk for any m1 > m and any f. E N. Finally, choose k1 > k and m1 > m according to {2) but w. r. t. n1• Put C(/) = CJrm' . Then we have, for all (�, TJ) , (�1, TJ1) in C(/) : Vn
TJ f 2: b1o = TJ1 f2: b1o => �f �an = �1 f2: an TJ r 2: blo' i= TJ t2: blo' => � t 2:an' i= � t2: a n' I
I
}•
{3)
Note that *S = dom19 = Ut eF X(/), where X(/) = dom C(f). Thus by F1 � F such that *S = Ut eF' X(/). Let us show that all sets X(/) are too small for a finite union of them to cover *S. Call an internal set X � *S small iff (* ) there is a number z E *N ' N such that, for any internal map a E int2 •N , [O ,z } , the set Xa = {� E X : �f2: = a} satisfies 2- z#Xa � 0. z Clearly *S is not a union of finitely many small internal sets. To get a con tradiction, it remains to show that any set X(/) is small, with z = an' in the notation above. (Note that an' depends on /, of course.) Take any (�, TJ) E C(/) and let a = �t > an , , r = TJ t > bk , By {3), each (e, TJ1) E C{ /) with e t> a = a satisfies -TJ1 t > b = T. Divide the domain "CJ! = {TJ1 E *S : TJ1 f > blo , = rl into subsets "CJ!w = {TJ1 E "CJ! : TJ1 f [bk, bk' ) = w}, where w E int2[b�o ,b k' > , totally 261o' - b1o of the sets "CJ!w. For any such "CJ!w , the set gjw = { �� : 3 TJ1 E "CJ!w (�1, TJ1) E C(/)} contains at most 2 a n elements by the first implication in {3). Thus X(/)a = {e E X(/) : �� t > a n , = a} contains at most 2a n +b�o, -b�o elements of the set X(/), which is les� than 2an' -l for any f. E N. We conclude that X(/) is small, as required. Part 2. Suppose that rate U � rate V. In this case it does not take much effort to redefine the sequences {an } , { bk } , cofinal in resp. log U, log V, so that an + l - an � bn+ l - bn for all n E N. By Robinson 's lemma (Theorem 2.2. 12), there exist a number h E *N ' N and internal extensions Saturation there is a finite set
•
I
I
9. 7 Equivalence relations and quotients
385
{a,} ,� h and {b,},� h of sequences {a n }ne N and {bn}ne N , both being in creasing hyperfinite sequences satisfying a,+1 - a, � bv+ l - b, for all v < h. Now to prove that R1og u �B R1og v we define a Borel reduction ah . That such a map
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V. K A N O V E I
·
M . REEKEN
Nonstandard Analys is, Axiomat ically
The book is devoted to nonstandard set theories that se rve as foundat ional basis for nonstand ard mathematics. Several popular and some less known nonstandard theories are considered, includ ing internal set theory 1 ST, Hrbacek set theory H ST� and others. The book p resents the basic st ruct ure of the set universe of these theories and methods to effect ively develop
"applied)) nonstandard analysis, metamathemat ical proper t ies and interrelat ions of these nonstandard theories between each other and with ZFC and some variants of ZFC, foundational
problems of the theories, in cluding the problem of external sets and the Power Set problem, and methods of their solution. The book is oriented towards a reader having some experience in foun dations (set theory, model th eory) and in n on s t anda rd
analysis.
ISSN
1439-7382
ISBN
3- 540-22243-X
9
II 1 1 1 1 1 111 1
7 8 3 540 2 2 2 439
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