Mathematical Logic in the 20th Century

MATHEMATICAL LOGIC IN THE 20TH CENTURY

HHHI-f

MATHEMATICAL LOGIC IN THE 20TH CENTURY

— —4

Gerald

E.

Sacks

P S S SINGAPORE UNIVERSITY PRESS \jy%/

NATIONAL UNIVERSITY OF SINGAPORE

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vii Introduction Brevity is the soul. H. M. First come the disclaimers, then the rules for selecting the papers, the exceptions to the rules, the justifications of the exceptions, and finally some brief remarks on the papers. (But before all that, the point of it all. The original papers are studied in the hope of recovering early ideas lost in later expositions. Proofs are rare, but the ideas used in proofs are rarer still.) The title of this volume is too broad. Almost all of the papers belong to the second half of the twentieth century. The last decade of the twentieth century is lightly represented. Only so much can be forced into one volume. The first half of the last century is well represented elsewhere; now is too soon to reach conclusions about its final decade. I have not read all the logic of the last century — far from it. And only a fraction of what was read was understood. The choices made were personal in nature. Who knows what "personal" means? The selection was certainly not based on an Olympian view of mathematical logic derived from a long and scholarly life of pondering the subject. Perhaps the choices cohere, if only because it is hard to see how it could be otherwise. The selection rules were: Rl. R2. R3. R4.

No papers from before World War II. No long papers. At most one paper by any author. The paper was (and is) intellectually exciting then (and now).

The first three rules are justified above. They proved difficult to follow. Godel's pre-war National Academy paper on the generalized continuum hypothesis is short, readable, more illuminating than his subsequent Princeton Mathematics orange book, and probably closer to his early thinking. His Academy paper, but not his orange book, mentions Russell's Axiom of Reducibility as a source of inspiration. The papers by Kleene and Tarski are much too long for this volumn, but Kleene is the father of recursion theory and Tarski of model theory. Why this particular paper by Kleene? Two reasons. He lifts the concepts of classical recursion theory to objects of finite type, and he shows that sets of non-negative integers are hyperarithmetic if and only if they are recursive in 2E, the type 2 object corresponding to the number quantifier. The title of Tarski's paper speaks for itself. Rule R4 was not violated, possibly a necessary fact. Cohen has two papers intended for this volume, but they are in fact the two halves of one paper. For this work, he received the Fields Medal, the highest international award in mathematics. Godel's paper transformed set theory into a subject that welcomes a wide range of mathematical ideas. His use of the downward Skolem Lowenheim theorem inside

viii L is the beginning of fine structure theory. His paper combined with Cohen's puts Cantor's continuum problem outside the conventional realms (ZFC) of set theory. Cohen's paper introduced the method of forcing, an essential technique with applications throughout logic. Forcing has unconscious precursors in recursion theory, for example the construction of a minimal Turing degree in Spector's paper. Silver's paper proved (in ZFC) a new theorem of cardinal arithmetic at a time when such an outcome was thought unlikely because of the Godel and Cohen results. He applied some ideas about ultrafilters to show: if the generalized continuum hypothesis holds below a singular cardinal K of uncountable cofmality, then it holds at K. This line of thought led to Shelah's pcf theory [II], which yields estimates on the size of the power set of a singular cardinal of countable confinality. ([Im] is the mth item in the References at the end of this introduction. All other papers mentioned are from the Contents list for the volume.) Choosing a paper by Shelah was a daunting task because of the large number of his contributions to model theory and set theory and the limit imposed by rule R2. His 1969 paper on stability can be seen as the beginning of his sweeping transformation of model theory. The notion of w-stability originated in Morley's proof that a countable theory categorical in some uncountable power is categorical in all uncountable powers. Morley's paper, building on Vaught's earlier paper, was the beginning of modern model theory. Vaught's paper went beyond immediate applications of compactness and stressed the notion of element type. For example he showed that the number of countable models up to isomorphism of a complete countable theory could not be two. Jensen's covering theorem, in his paper with Devlin, makes a connection between sets of ordinals in V and L with the help of fine structure theory and the work of Silver [12] and Solovay [13] on OK The choice of Friedberg's paper on recursive enumeration came about as follows. He in [14] and Mucnik independently solved Post's problem by introducing the priority method, a technique that dominates classical recursion theory to this day. By choosing Mucnik's version, I could satisfy rule R3 and still include Friedberg's construction of a maximal recursively enumerable set, a result that ignited interest in the lattice of recursively enumerable set under inclusion modulo finite sets. The lattice was initially studied in Post's paper on recursively enumerable sets and their decision problems. The lasting influence of Post's paper entitles him to be called the co-father, if there is such a thing, with Kleene of recursion theory. Soare's paper showed any two maximal sets are automorphic. His result is the reason that the lattice continues to be of interest. Post's paper established the legitimacy of an intuitive approach to recursion theory: less equations and more words. Friedberg's paper is any early example of the intuitive style. Spector's paper adheres to Kleene's formal style, only because it was extracted from Spector's thesis supervised by Kleene. Lachlan's paper introduced the so-called (but not by him) monstrous injury method, close to the final stage in the development of the Friedberg-Mucnik priority method. Moschovakis's paper found a nearly paradoxical role for divergence in recursion theory and led to constructions of recursively enumerable sets in higher recursion theory in which divergence witnesses played as big a part as convergence witnesses.

ix

Matijasevic's paper on the unsolvability of the Diophantine problem has historical antecedents as old as any in mathematics. The underrepresentation of proof theory in this volume indicates nothing more than my own confusion over the subject. The most striking proof theorist I have met is Girard. His paper was chosen for its brevity and as an example of his unique mode of thought. In it he discusses his concept of dilator. Kreisel, another proof theorist with his own mode of thought, is also included. His paper is a mixture of recursion theory, model theory, proof theory and other subjects hard to put a name to. He presents a compactness theorem for w-logic based on his insight that generalizing the notion of finite is the key to extending various results in model theory and recursion theory. In this compactness theorem "hyperarithmetic" is the generalization of "finite". Robinson's paper on non-standard analysis is a model theorist's way of making sense out of infinitesimals. Godel thought it was of historic importance. Wilkie's paper solves a long standing problem of Tarksi on the first order theory of the reals with the exponential function added. The work of Zil'ber and Hruschovski bring ideas of geometry and stability to bear on model theory. Zil'ber's paper was chosen as a brief example of his approach, and Hruschovski's paper as a prime application of model theory to number theory. H. Friedman's paper shows that Borel Determinateness (BD) cannot be proved without invoking objects of arbitrarily high countable rank despite the fact that BD is about Borel sets of reals, objects of rank 1. Later Martin [15] proved BD by means of an induction that trades decreases in rank of Borel sets for increases in rank of objects. Solovay's paper assumes the consistency of "there exists an inaccessible cardinal" and then demonstrates the consistency of "every set of reals is Lebesgue measurable and countable dependent choice". Later Shelah [16] proved the converse. Scott's paper showed the existence of a measurable cardinal implies the existence of a non-constructible set. This result, and its proof via ultrapowers, broke open the study of large cardinals. Martin's paper used a measurable cardinal to establish the determinacy of analytic games. His argument needed only the sort of indiscernibles provided by O". Later the converse was shown by Harrington [17]. (Thus Cfi is equivalent to lightface nj determinacy.) Martin's result eventually led to complex connections between determinateness and large cardinals obtained by Woodin, whose paper in this volume is a brief example of his unique insight. Shoenfield's £2 absoluteness result is a personal favorite. It has applications throughout logic. One example is the Slaman-Woodin proof [18] of the definability of the double Turing jump. My thanks to those who suggested papers for this volume. They insist on remaining anonymous. Cambridge, Massachusetts December 21, 2002

X

References for the Introduction [II] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York 1994. [12] J. Silver, Some applications of model theory in set theory, Ann. Math. Logic 3 (1971) no. 1, 45-110. [13] R. Solovay, A nonconstructible A3 set of integers, Trans. Amer. Math. Soc. 127 (1967), 50-75. [14] R. Friedberg, Two recursively enumerable sets of incomparable degrees of unsolvability, Proc. Nat. Acad. Sci. USA 43 (1957), 236-238. [15] D. A. Martin, Borel determinacy, Ann. of Math., Ser. 2, 102 (1975) no. 2, 363-371. [16] S. Shelah, Can you take Solovay's inaccessible away?, Israel J. Math. 48 (1984), no. 1, 1-47. [17] L. Harrington, Analytic determinacy and O", J. Symbolic Logic 43 (1987), no. 4, 685-693. [18] T. Slaman and H. Woodin, Definability in degree structures, forthcoming.

xi

Contents

Introduction

vii

The Independence of the Continuum Hypothesis Cohen, Paul J.

1

The Independence of the Continuum Hypothesis II Cohen, Paul J.

7

Marginalia to a Theorem of Silver Devlin, K. I. and Jensen, R. B. Three Theorems on Recursive Enumeration. I. Decomposition. II. Maximal Set. III. Enumeration without Duplication Friedberg, Richard M.

13

41

Higher Set Theory and Mathematical Practice Friedman, Harvey M.

49

Introduction to II^-Logic Girard, Jean-Yves

82

Consistency-Proof for the Generalized Continuum-Hypothesis Godel, Kurt

108

The Mordell-Lang Conjecture for Function Fields Hrushovski, Ehud

113

Model-Theoretic Invariants: Applications to Recursive and Hyperarithmetic Operations Kreisel, G. Recursive Functionals and Quantifiers of Finite Types I Kleene, S. C. A Recursively Enumerable Degree which will not Split over all Lesser Ones Lachlan, A. H. Measurable Cardinals and Analytic Games Martin, Donald A.

137 153

205 264

xii Enumerable Sets are Diophantine Matijasevic, Ju. V.

269

Categoricity in Power Morley, Michael

274

Hyperanalytic Predicates Moschovakis, Y. N.

299

Solution of Post's Reduction Problem and Some Other Problems of the Theory of Algorithms Mucnik, A. A.

333

Recursively Enumerable Sets of Positive Integers and Their Decision Problems Post, Emil L.

352

Non-Standard Analysis Robinson, Abraham

385

The Recursively Enumerable Degrees are Dense Sacks, Gerald E.

394

Measurable Cardinals and Constructible Sets Scott, Dana

407

Stable Theories Shelah, S.

411

The Problem of Predicativity Shoenfield, J. R.

427

On the Singular Cardinals Problem Silver, Jack

435

Automorphisms of the Lattice of Recursively Enumerable Sets Part I: Maximal Sets Soare, Robert

439

A Model of Set-Theory in which Every Set of Reals is Lebesgue Measurable Solovay, Robert M.

480

On Degrees of Recursive Unsolvability Spector, Clifford

536

A Decision Method for Elementary Algebra and Geometry Tarski, Alfred

548

xiii Denumerable Models of Complete Theories Vaught, R. L. Model Completeness Results for Expansions of the Ordered Field of Real Numbers by Restricted Pfaffian Functions and the Exponential Function Wilkie, A. J. Supercompact Cardinals, Sets of Reals, and Weakly Homogeneous Trees Woodin, W. Hugh

609

628

672

Structural Properties of Models of Ni-Categorical Theories Zil'ber, B. I.

677

Permissions

691

1

VOL. 50, 1963

MATHEMATICS: P. J. COHEN

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THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS BY PAUL J. COHEN* DEPARTMENT OF MATHEMATICS, STANFORD UNIVERSITY

Communicated by Kurt Godel, September 30, 1963

This is the first of two notes in which we outline a proof of the fact that the Continuum Hypothesis cannot be derived from the other axioms of set theory, including the Axiom of Choice. Since Godel3 has shown that the Continuum Hypothesis is consistent with these axioms, the independence of the hypothesis is thus established. We shall work with the usual axioms for Zermelo-Fraenkel set theory,2 and by Z-F we shall denote these axioms without the Axiom of Choice, (but with the Axiom of Regularity). By a model for Z-F we shall always mean a collection of actual sets with the usual e-relation satisfying Z-F. We use the standard definitions3 for the set of integers w, ordinal, and cardinal numbers. THEOREM 1.

There are models for Z-F in which the following occur: (1) There is a set a, a C « such that a is not constructive in the sense of reference 3, yet the Axiom of Choice and the Generalized Continuum Hypothesis both hold. (2) The continuum (i.e., (P(o;) where (P means power set) has no well-ordering. (3) The Axiom of Choice holds, but & ^ 2No. (4) The Axiom of Choice for countable pairs of elements in (P((P(u>)) fails.

Only part 3 will be discussed in this paper. In parts 1 and 3 the universe is wellordered by a single definable relation. Note that 1 implies that there is no simple ordering of (P((P(w)). Since the Axiom of Constructibility implies the Generalized Continuum Hypothesis,3 and the latter implies the Axiom of Choice,5 Theorem 1 completely settles the question of the relative strength of these axioms. Before giving details, we sketch the intuitive ideas involved. The starting point is the realization1' ' that no formula n(.r) can be shown from the axioms of Z-F to have the property that the collection of all x satisfying it form a model foi Z-F in which the Axiom of Constructibility (V = L,3) fails. Thus, to find such models, it seems natural to strengthen Z-F by postulating the existence of a set which is a

2 1M-1

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P. J. COHKX

l'nor. N. A. S.

model for Z-F, thus giving us greater flexibility in constructing new models. (In the next paper we discuss how the question of independence, as distinct from that of models, can br handled entirely within Z-F.) The Lowenheim-Skolem theorem yields the existence of a countable model 9TC. Let N1; &>, etc., denote the corresponding cardinals in 971. Since 911 is countable, there exist distinct sets as C u, 0 < 5 < N2. Put V = \(ai: a y ) | 5 < 5'}. We form the model 31 "generated" from 9TC, o.j, and I7 and hope to prove that in 91 the continuum has cardinality at least X2. Of course, 91 will contain many new sets and, if the as are chosen indiscriminately, the set N2 (in 971) may become countable in 91. Rather than determine the as directly, we first list all the countably many possible propositions concerning them and decide in advance which are to be true. Only those properties which are true in a ''uniform" manner for "generic" subsets of o> in 9TC shall be true for the as in 91. For example, each as contains infinitely many primes, has no asymptotic density, etc. If the as are chosen in such a manner, no new information will be extracted from them in 91 which was not already contained in 371, so that, e.g., {^2 will remain the second uncountable cardinal. The primitive conditions n e as are neither generically true nor false, and hence must be treated separately. Only when given a finite set of such conditions will we be able to speak of properties possibly being forced to hold for "generic" sets. The precise definition of forcing will be given in Definition 6. From now on, let 911 be a fixed countable model for Z-F, satisfying V = L, such that x e 971 implies x c 3TC. If 9TI' is a countable model without this property, define ^ by transfinite induction on the rank of x, so that ^(x) = {y J z e 971', z e x, ~&(z) = y); the image 3TI of 9TI' under ^r is isomorphic to 3TC' with respect to e and satisfies our requirement. Thus, the ordinals in 971 are truly ordinals. Let T > 1 be a fixed ordinal in 3TC, KT the corresponding cardinal in 9TC, and let as, 0 < 5 < b$T be subsets of w, not necessarily in 971, V = {(as, a,y)\ S < 5'}. LEMMA 1. There exist unique functions j , K\, Kz, N, from ordinals to ordinals definable in 971 such that (1) j(a + 1) > j(a) and for all 0 such that j(a) + K j 3 < j ( a + 1) the map jS -*- (N(i3), Ki(@), Ki(ff)) is a 1-1 correspondence between all such /?, and the set of all triples (i, y, 8), 1 < i < 8, y < j(a), S < j(a). Furthermore, this map is orderpreserving if the triples are given the natural ordering S (Ref. 3, p. 36). (2) j(0) = 3NT + 1, j(a) = sup{j($) \ff < a} if a is a limit ordinal. (3) N(j(a)) = 0, N(j(a) + 1) = 9, Kt = Ofor these values. (4) N(a), Kt(a) are zero if a < 3NT. (5) / / 0 is as above, and N(p) = i, put J(i, Kx(fi), K2Q3), j(a)) = 0. Also put 1(P) = J(«)Definition 1: For a an ordinal in STU, define Fa by means of induction as follows: (1) Fa = a if a < co. (2) For u < a < 3KT, let Fa successively enumerate ait the unordered pairs (a5, as) in any standard manner (e.g., the ordering R on pairs of ordinals def. 7.813), (3) For a = 3« T , Fa = V. (4) For a > 3NT> if £,(«) = ff, K*(a) = y if N(a) = 0, Fa = {Fa,\a' < a}

if 1 < i *= N(a) , and a, in place of Fs + a + l, etc., if there is no danger of confusion. If N{a) = 9, then the set Fa is defined independently of as. We shall now examine statements concerning Fa before the as are actually determined, and thus the Fa for a while shall be considered as merely formal symbols. Definition 2: (1) x e y, x e Fa, Fa e x, Fa e Fp are formulas; (2) if

P, P' does not force Fa = Fp'. Again the symbol = is treated as before. (v) If a > /8, we reduce the case Fa t Fp to cases (in) and (it;) treated above. We say P forces F a « F 0 if for some 0' < fi, P forces i*> e F s and P forces ^ a = Fp> (i.e., (x)a(a; e F a ( = ) x e i^*) which is a statement of type (R and hence precedes Fa e Fp). We say P forces ~]FatFfi if for all 0' < 0 and P ' 3 P, P ' does not force both Fp' e Fp and Fp, = ^ a . The most important part of Definition 6 is I, the other parts are merely obvious derivatives of it. Definition 7: If a is an unlimited statement with r quantifiers, we define "P forces a" by induction on r. If r = 0, then a is a limited statement. If a = (x) h(x), P forces a, if for all P ' 3 P, and a, P' does not force 1 b(Fa). If a = Jx b(x), P forces a if for some a, P forces b(Fa). In the proofs of Lemmas 2, 3, 4, and 5, we keep the same well-ordering on limited statements as in Definition 6, and proceed by induction. LEMMA 2. P does not force a and ~~\ a, for any a and P. Proof: Let a be a limited statement with r quantifiers. If r > 0, and P forces both JaX b(x) and (x)a ~] b(x), then P must force b ^ ) for /? < a which means P cannot force (x)a ~\ b(x). Case II of Definition 6 will clearly follow from case III. Parts (i) and (ii) are trivial. If a is in part (Hi), then P forces a if and only if P

5 VOL. 50, 1963

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1147

forces a statement of lower rank and in this case the lemma follows by induction. In part (iv), if P forces Fa which means P can not force ~\ Fa e Fp. In part (v) if P forces Fa t Fp, for some 0' < 0, P forces Fp> e Fp and Fa = Fp> which again violates P forcing ~i Fa e Fp. If a is an unlimited statement, the lemma follows in the same manner by induction on the number of quantifiers. LEMMA 3. If P forces a and P' ZD P, then P' forces a. Proof by induction as in Lemma 2. LEMMA 4. For any statement a and condition P, there is P' 3 P such that either P' forces a or P' forces ~~\ a. Proof: Let a be a limited statement with r quantifiers. If r > 0 and P does not force a = (x)a b(x), then for some P' Z) P, P' forces ~\ b(Fp), 0 < a, which means P' forces ~\a. If r — 0, we may restrict ourselves to III, for if we enumerate the components of a, by defining a finite sequence Pn, PB = P and P , + i D P , we may successively force each component or its negation so that finally either a or 1 o is forced. Again, cases (i) and (ii) are trivially disposed of. Case (in) is handled by induction as before. If a = Fa e Fp is in case (ID) then if P does not force ~\ a, for some P' => P and 0' < p, N(p') = 9, P' forces Fa = Fe. so P' forces a. If a == Fa e Fp is in case (v) if P does not force ~\ a, then for some P' Z) P, /3' < 0, P' forces Fp e Fp and Ffi- = Fa> hence P' forces a. Unlimited statements are handled as before. Definition 8: Enumerate all statements an, both limited and unlimited, and all ordinals an in 3E. Define Pzn as the first extension of Pin _ \ which forces either an or ~l an. Define P 2n + i as the first extension of Pin which has the property that it forces Fp t F«n where 0 is the least possible ordinal for which there exists such an extension of P2re, whereas if no such /3 exists, put P2« + i = PinThe sequence Pn is not definable in 3TC. Since all statements of the form n e as are enumerated, Pn clearly approach in an obvious sense, sets as of integers. With this choice of ae, let 91 be defined as the set of all Fa defined by Definition 1. LEMMA 5. All statements in 91 which are forced by some Pn are true in 91 and conversely. Proof: Let a be a limited statement with r quantifiers. If r > 0, then if P,, forces (x)a b(x), if /3 < a, then some Pm must force b(Fs) since no Pm can force ~| b(Fp). By induction we have that h{F0) holds, so that (x)a b(x) holds in 31. If Pn forces Jax b(x), for some /3 < a, Pn forces b(Ffi) so by induction b(F?) holds and hence laX b(x) holds in 91. Case I I will clearly follow from case I I I and (i) and (ii) arctrivial. If a is Fa e Fp or ~1 Fa e FB in case (in) then if Pn forces a, Pn forces precisely the statement which because of the definition of Fp is equivalent to a. In case (iv) if Pn forces Fa e Fp, for some /3' < 0, .V(j3') = 9, l\ forces F^' = F«, which therefore holds by induction in %.. If Pn forces ~| Fa t Fp, then for each p' < Ii, N(P') = 9, Fa = F 3 / is not forced by any P m so some P ra must force Fa ^ /'V' which proves ~| Fa t F0 holds in 31. Similarly for case (v) and for unlimited statements. Since every statement or its negation is forced eventually, the converse is also true. Lemma 5 is the justification of the definition of forcing since wo can now throw back questions about ill to questions about forcing which can bo formulated in 0??.

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PROC.

N. A. S.

In the next paper, we shall prove that, ill i^ a model for Z-F in which part 3 of Theorem 1 holds. * The author is a fellow of the Alfred P. Sloan Foundation. 1 Cohen, P. J., "A minimal model forset theory," Bull. Anier. Math. Soc, 69, 537-540 (1903). 2 Fraenkel, A., and Y. Bar-Hillel, Foundations of Set Theory (1958). 3 Godel, K., The Consistency of the Continuum Hypothesis (Princeton University Press, 1940). 4 Shepherdson, J. C , "Inner models for set theory," J. Symb. Logic, 17, 225-237 (1957). 6 Sierpinski, W., "L'hypothese gcneralisee du continu et l'axiome du ehoix," Fund. Math., 34 1-5(1947)

7

VOL. 51, 1964

MATHEMATICS:

THE INDEPENDENCE

P. J. COHEN

OF THE CONTINUUM HYPOTHESIS,

105

II*

BY PAUL J. COHEN1 DEPARTMENT OP MATHEMATICS, STANFORD UNIVERSITY

Communicated by Kurt Godel, November 27, 1963

This paper is a continuation of reference 1, in which we began a proof of the fact that the Continuum Hypothesis cannot be derived from the other axioms of set theory, including the Axiom of Choice. We use the same notation as employed in reference 1. THEOREM 2.

91 is a model for Z-F set theory.

The proof will require several lemmas. The first two lemmas express the princi-

8 106

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PROC. N. A. S.

pie that forcing is a notion which is formalizable in the original model 9TC. LEMMA 6. There is an enumeration aa of all limited statements by means of tlie ordinal numbers of 3TC, such that the usual formal operations -performed on statements are expressible by means of definable functions in 3TC of the indices a, for example, forming negations, conjunctions, replacing variables by partimdar sets, etc. Furthermore, the ordering corresponds to the definition of forcing given by trans/mile induction in Definition 6. LEMMA 7. Let o.{x,y) be a fixed unlimited statement containing two unbound variables x and y. The relation $a(P,«,/3) which says that ]' forces (i(Fa,Fp) and /3 is the least such ordinal, is definable in 3TC. This follows from the fact that using Lemma 6 the relation "P forces aa" can be formalized in Z-F as a statement about P and a. A given unlimited statement can also be handled since, after a finite number of replacements of variables, it is reduced to a limited statement. Definition 9: For a(x,y) as above, put To(a) = sup[/?|3 P,oti < a, $ 0 (P,ai,/8)}. LEMMA 8. Let a(x,y) be a fixed unlimited statement, a an ordinal. For each a' < a either there is no Fp such that a(Fa',F$) or such an F0 exists with /8 < r a ( a ) . Proof: If ,8 is the least ordinal such that a(Fa',Ffi), then a(Fa>,Fp) must be forced by some Pn which clearly implies (3 < ra(a). LEMMA 9. Let &(x,y) be an unlimited statement of the form QiXxQ&i,

• • -, Qnxnb(x,y,xh

. . . , x n)

where b has no quantifiers and Qt are either existential or universal quantifiers. In 31, assume <X defines y as a single-valued function of x. Then for each a tlicrc exist ordinals 7o, . . . , 7 r e such that for x e Ta, there exist y t Fyo such (hat a(x,y) and for (x,y) in Ta X Tyo, the statement a(x,y) holds if and only ifa(x,y) holds where a is the statement formed by restricting the quantifiers Qt in h to range over Fyi. Proof: Lemma 8 implies the existence of 70 such that for x t Ta, there is a y e Fyo such that a(x,y). Define yk by induction as follows: let gk(x,y,xu . . ., xk-\,z) be the condition (i) if Qk is universal, ~Qk+ixk+i,

(ii)

• • ., Qnx,,b(x,ij,xi,

. . ., xk_hz,xk+i

. . . xn)

or

if Qk is existential, Qk+ixk+x,

. . ., Qnxnb(x,y,Xi,

. . ., xk_hz,xk^l

. . . xn).

L e m m a 8 implies t h a t for some yk, for all {x,y,x,, . . ., xk^) e Ta X Fyo X . . . X Fyk_u either no z exists such t h a t gk{x,y,xu . . ., xk_hz) or t h e r e is such a z e Fyt. This clearly implies t h e lemma.

LEMMA 10. The Axiom of Replacement holds in 31. Proof: If a(x,y) defines y as a single-valued function of x in 31, then for any a if D = {x\jz, z t Fa& a(z,x)} then by Lemma 9, D is denned by a condition in which all variables are restricted to lie in fixed sets Fyi, which by the definition of the sets Fa implies that D is a set in 31. The only other axiom to verify which is nontrivial, is the Axiom of the Power Set. The proof we give follows closely the method in reference 2 used to prove that V = L implies the Continuum Hypothesis.

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LEMMA 11. Let W be a set in 3TC, consisting of conditions P, such that if Pi and Pt belong to W, then P\ U Pi is not an admissible condition {i.e., contains a contradiction). Then iV is a countable set (in 3TC). Proof: Define sequences nk and Pj as follows. Put n,i = 1 and P\ the first P in IT. (We assume the P are well-ordered.) If nk and P} for j < nk are defined, put Rk equal to the sot of till conditions n e as or ~ n e as such that they or their negations are contained in .some Pitj < nk. Let Pt, nk < i < nk+i, be finitely many P in W, such that for all I' in W, 3j, nk < j < nk+i and P and P , have precisely the same intersection with l{,:. This is possible since Rk is a finite set. We claim that W consists only of the I'). For if P t W, then since P is a finite set of conditions, and Rk Q /i\+i, tliere exists a k such that P fi Rk = P D fit+i. Let nk < j < nk+i, such that P , (1 / C /("t+i, / ' U Pj is an admissible condition, which contradicts the hypothesis. Definition 10: Put C(P,a) = ft if P forces ^ « P« and for P ' 3 P, 7 < 0, P ' docs not force /(1T e P^. If no such /3 exists, put C(P,a) = 0. The function C is definable in 9TC, by virtue of the general principle contained in Lemma G. LEMMA. 12. For any a, there are only couniably many (in 9TC) /? such that for some I', C(P,a) = p. Proof: For each such /3, pick one P such that C(P,a) = /3. Then the set of all such P must be countable by Lemma 10. LEMMA VS. Let S be an infinite set of ordinals in 31!. There exists a set S' of onlinalx, S' 3 .S', S' = S such that S' is closed under J(i,a,/3,y), Ki(a), C(P,a), I (a), far all I' and a, ft, 7 e S. Also a t S' implies a + 1 e S'. The statement S' = S, above, means that with respect to 3TC, the sets S and S' arc of the same cardinality. LEMMA 14. Let S be a set of ordinals closed under the operations in Lemma IS, and such that if a < 3KT, a e S. Then there is a map g mapping S 1-1 onto an initial segment of ordinals which preserves J, Kt, I, N, and such that if N(a) = 0 (or 9), g(a) = ft is the first ordinal such that N(ft) = 0 (or 9) and ft is greater than g(a') for a' < a. Also, g is the identity for a < S^ r . Proof: S and g in the lemma refer to sets in the model 3K. We define g by transfinite induction. For a < 3^ T , let g be the identity. If g is defined for all ft in S less than a, UI(a) = a 3 (i.e., N(a) = 0), putg(a) = sap{g(ft)\ft < aaud/3 e S]. If / ( a ) = ft < a, then if N(a) = 9 (i.e., a = ft + 1), put g(a) = g(ft) + 1. If i = N(u), 1 < i < 8, put g(a) = J(i,g(Ki(a)), g(K*(a)), g(ft)). One can now show by induction that if a e S, N(a) = 0, g maps the set of all ft < a onto an initial segment. The lemma then easily follows. LEMMA 15. If we put G(Fa) = F0(a) for a in S, then G is an isomorphism with respect to e of Ax = {Fa\a e S] onto A2 = {Fg(a)\a t S]. Proof: This follows by induction on a, in the same way as in 12.6 of reference 2. Observe that in examining the operations JF4 and EF5 we need the fact that if Fae Ai and is not empty, then it has a member in Ai preceding it. This is true since S is closed under C(P,a), and C(P,a) for some P is the smallest ft for which Fpt Fa, if Fa^ . LEMMA 16. / / Fp C Fa, then for some y, Fp = Fy, where y < 3 + X r in 3R.

10 108

MATHEMATICS.

P. J. COHEN

PROC. N. A. S.

Proof: Let S contain all 8 < a, all 8 < 3 ^ r and /3, and be closed under the operations in Lemma 13. Let g be the corresponding isomorphism. Then clearly g(S) = 8 if 8 < a. Thus, by Lemma 15, if we put y = g(/3), since Fp CZ Fa! Fy = F0. Since g maps <S onto an initial segment, "y < S and so the lemma is proved. LEMMA 17. The Axiom of the Power Set holds in 91. Proof: Since every subset of Fa is contained in Fp, where /3 is the first ordinal such that iV(/3) = 0 and ^ > a + XT, it is clear that the power set of Fa occurs in 91. This completes the proof that 91 is a model, the other axioms being trivially verified. Since rank Ff< a, 31 contains no new ordinals. LEMMA 18. / / N(a) = N(0) = 9, and Fa > Fp in 9ft, then Fa > Fp in 91. Proof: The point of this lemma is that ordinals do not change their relative cardinality in the model 91. The added complications in the definition of forcing due to N(a) — 9 are compensated for in the proof of this lemma, in that as a runs through the ordinals with N(a) = 9, Fa runs through the ordinals of 311 in a manner independent of the sequence Pn. More exactly, the map a -*• Fa is an orderpreserving map of the ordinals a, N(a) = 9, onto all the ordinals of 311. Thus assume that some element in 91 defines a relation 6+.

Since forcing is the natural method for producing independence results, set theorists have concentrated on a more specific form of the problem: Given a transitive model M of ZP +

GCH, can a positive solution be ob-

tained by forcing over M with a set of conditions IP ? This approach suggests a number of related problems: Is there a

IP which collapses 8 + to

6? Is there a IP which makes an inaccessible cardinal singular? Until very recently, there was a widespread assumption among set theorists that such sets of conditions do exist and"merely awaited discovery. Then Silver challenged this assumption by proving it false. Specifically, Silver proved - in ZPC - that if the continuum hypothesis holds below a singular cardinal f$ of uncountable cofinality, then it holds at 6. Thus, in many important cases, not only the narrower forcing problem but the general problem itself has a negative solution.

Much of the effort to produce a positive forcing solution centered on the attempt to exploit the properties of special ground models - either L or models containing large cardinals. The latter approach met with some success: Prikry, fr. ins., showed that a measurable cardinal can be turned into an ID cofinal cardinal. Magidor, starting with an elephantine cardinal, produced a model in which 2

n

< oi for n < ui and 2

w

> is + ..

Jensen's efforts to produce a positive solution over L led to total

.

14 116

K. Devlin & R. Jensen

failure. Silver's work then led him to consider the problem from a new perspective. He discovered that the statement " 0 * does not exist" (henceforth abbreviated as -i 0 * ) implies a negative solution to all cases of the singular cardinals problem. But then there cannot be a positive forcing solution over L, since every generic extension of L by a set of conditions satisfies -i 0

Throughout this paper we assume ZFC. Our main theorem says, in effect, that if -i 0

, then the "essential structure" of cardinalities and con-

finalities in L is retained in Theorem 1.

Assume -i 0

V.

. Let X be an uncountable set of ordinals. Then

there is a constructible set Y s.t.

X 2 by co. in Corollary 2. The following corollary establishes a totally negative solution of the singular cardinals problem over L. Corollary 3. (a)

Assume -i 0

. Let B be a singular cardinal. Then

B is singular in L

(b) e + = B + L (c)

If A c g s.t. Hg = Lg[A], then iP(B) c L[A].

(d)

cf(B) s Y < B

(e)

Let 0 = 2 ^ . Then R

—*• B Y = 2 Y • B +

feif\/Y « is suitable iff either J T ^— (There is a largest cardinal)

or else there are arbitrarily large y < x s.t. cf(y) > a> and J T

[=•= ( Y is

regular). § 0 Theorem 1 reduces to the statement: Lemma 1.

Let T i u>2 be a suitable cardinal in L. Let X c x be cofinal

in t s.t, I < T . Then there is Y 3 X s.t.

Y 6 L

and Y < x.

We first show that Lemma 1 implies § 0 Theorem 1. Suppose not. Let X be an uncountable set of ordinals for which the conclusion of §0 Theorem 1 fails. Choose x = lub(X) minimal for such X. Then X < 7, since otherwise the conclusion of § 0 Theorem 1 would hold with Y = x. Hence x > (»„. Now suppose that the conclusion of Lemma 1 held. There would then be Z € L s.t. X c Z

and t < T . Let p = z

and let f : p -


the

zn

standard code and the Z n standard parameter of a. We

recall the following facts: (1)

p n s the largest p s.t. <J , A) is amenable for all A € En(Ja) n 9 ( J p ) .

(2)

R c Jpn

is S n (J a ) iff R is

Z^J^-1

, A^ ).

(5) A° » P ° = 0 . (4) Let n * 1 and let h be the canonical S^ Skolem function for <J n-1 , R11'1). Pa 01

h"(w x J

Then p n is the least p s.t. J n-1 = p™ n a x (p}) for some p 6 J n and p a is the <j - least such p.

(5) R c J n is I j W n - i , A^~ ) in the parameter p^ iff R is rud in a a <J n , A > (i.e. R is the intersection of J n with a class rudimena a tary in A o ) '

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K. D e v l i n & R. Jensen

L e t IT : <J— , A) —>•_ < J \ n , A ) ( i 2 0 ) . Then t h e r e i s a P **.: P a 1 a o » ^ s,t. s.t.

(T = p— , A" = A—. M o r e o v e r , J a and i r ( p - )

TT : J - — > z

unique

t h e r e i s a u n i q u e if z> IT

= p (j

£

n).

n+i All of these facts are established in [Dev] and [PS]. The next result, though not explicit in our reference articles, does indead follow easily from the above facts. Def 1

Let a £ 6, 0 & n s u>: »a is a E

•'••'•

—

there is no

sn(Jg)

cardinal {!„ regular) in Jo iff ri

'" •' ™

'

n

• '™~

——•

p

function mapping a subset of some y < ua onto (cofinal-

ly into) ma. a is a cardinal (regular) in Jg iff there is no f 6 Jg mapping a y < wet onto (cofinally into) ua. If a is a cardinal in J o and a 6 Ja s.t. a c J . p

p

cr

then <J a , a) is amenable.

Clearly, being a cardinal (regular) in J g is the same as being a ZQ cardinal (regular) in Jg.

Lemma 1.

Let n i 1, a s (3.

(i) If iiia is a I . cardinal but not a Z cardinal in J o , then n n 1 n p o < a s p_ . Moreover up. is the least y < oa s.t. there is 1 (J o ) p S B n p map of a subset of y onto a>a. (ii) If p. < a s p P

and a is regular in J.n-1 , then cf(ua) = P

PQ

P

cf(o ) p n " 1 ). Proof. Q

(i)

Pg = B i a- Using C O , (2) and the fact that for any p there is a

A^(J ) map of up onto J , we get: p g a a for i < n (by induction on i ) . Hence Pg

2 • a v s o , since f

U f =f •

c f . Finally, sup a

= wo since

QED

V Carrying the proof of Lemma 1 (ii) a step further, we get the following rather technical lemma which will be of service to us in § 5.

Lemma 2.

n-l n Let p Q a o > p. where a>a is regular in J o . Let X = cf ( w ) .

Then there is a sequence ir, p i a, A c J— s.t. p g £ rng(ir) and

20

122

(b) (c) (d)

K. Devlin & R. Jensen _ n-1 _ n-1 There is a unique F s . t , p = p^- , A = A^ . n _ p^- < a . n n n n If T ( P ^ ) = Pg > then ~(p^-) = P g .

Proof. We first prove the existence part of (a). Let p, A, p, Y , ) •*

>- Vi < a) T-(4,,«i,x>,<x>>).

But the last equivalence is expressible as a H- statement in <J—, A") (since Vi < to T"(,5>) = P

serving and rng(3') n J

= rng(c).

hence by the fact that a is Z. pre-

Hence p— ^ < p .

Contradiction!

QED (Fact 6) Facts it, 5, 6 immediately give:

Fact 7.

A = An

R

All that remains to be proved is Fact 8.