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A , > - . . > A , andO 0, W Ais a principal number for addition. 10.1.8 LEMMA. If A€&'( +) then A = W" for some n, or for all n, W"< A . PROOF.If A E ~+)( then A > 1 and hence by lemma 6.3.1, l
a
w
=
W5A.
Ch. 101
UNIQUENESS RESULTS
125
If A =I= W, then W O are principal numbers for addition. By lemma 10.1.8, these are the only principal numbers for addition with classical ordinals , ( 1 , 1 > > T s(A). Next we define E(A) to be the smallest set E of ordered pairs of natural numbers satisfying 1)-3) below 1) If (x, Y)E t(A), then
2) Tf113x,2E.-.2Ex,, where r ~ 1 , m 1 , . . . , m r andeither ~1 r > l or m, > 1 or x1 is not of the form e
(" ') and analogous conditions hold for
Y ~ , . . . , Y ~ , ~ , ~ ~ , then ...,~,, (e
(m , ".... m,O e (i: ...n,'">) O E x1
3) If x
xrl)y
=e
(:
E
"'
"'":'> ... n 0
if, and only if ,
and (x, X ) E E and y = e
and ( y , Y ) E E ,then ( x , , y ) ~ Efor 1 I i l r , otherwise.
( x , Y ) E E if ( y , x) E E
Clearly E (A) and C'E (A) are uniformly primitive recursive in A. 11.1.3 THEOREM. If A is a well-ordering of type a, then E(A) is a wellordering of type E,. PROOF. Let cp be the map from E, = { fi :/3 <E,} into C'E ( A ) given by
Ch. 111
133
E-NUMBERS
where y < a, lxlA = y ,
/3 is not an &-number,o or 0 , cp (pi) = xi and /31 p,. > * * a >
We prove by transfinite induction that cp is well-defined and maps E, one-one onto C'E(A) in an order preserving way; from this the theorem follows at once. cp is well-defined since if B<E, then either (i) exactly one of p=O, p=o, f?=E,, for some unique y ...> /Ir. (Note that all the pi are <j? in this case.) That cp is one-one is clear from the uniqueness of the Cantor normal form. Since every ordinal < E , is expressible in the forms given above it follows that cp is onto and the order preserving property of cp is obvious from definition 11.1.2 and the observation that dl-ml
+ . . a +
[email protected]*rn,I*
ifweput p ( x ) = q e
THEOREM. A- BeE(A)-E(B), PROOF. Similar to the above: left to the reader.
11.1.6
This theorem justifies the following definition.
Ch. 111
135
E-NUMBERS
11.1.7 DEFINITION. E(A)=COT(E(A)) for any AEA. A co-ordinal of the form E(A) is said to be an E-number. 11.1.8 COROLLARY. A sBoE(A)sE(B). PROOF. Immediate from lemma 11.1.5.
11.2 The following is the main theorem of this chapter. 11.2.1 THEOREM. The following statements are equivalent: (i) X is an infinite principal number for exponentiation, (ii) P=x, (iii) X = w or w'=x, (iv) X = W or X is an E-number. PROOF. (i)=.(ii) follows from the definition of principal numbers. For (ii)*(iii) we need a lemma. 11.2.2 LEMMA. 2'=X*X= WD for some D>O. We first sketch the idea of the proof. Since p :2' N X for some p, we can represent each 1x1 for XEC'Xin the form 1x1 = 21x11
+...+ p.1 + 2"' + ...+ 2"r,
where rnl>-.'>m, are the (only) finite exponents. Given such an expression then 1x1 1 is obtained by adding 2' and reducing the result so that we again get a unique expression. Clearly this addition can only affect the finite exponents and so the problem is to determine explicitly and effectively how "far in" the 2' has an effect on the exponents. The calculation of t = t ( x ) , below, does just this. The function J/ enumerates the elements of C'X corresponding to the finite ordinals. Using these two functions it easily follows that X N W - D where D is obtained from the numbers which do not represent successor numbers in X. PROOF. Let XEX, then by hypothesis there exists p such that
+
p:2%
x.
Let J/ be the partial recursive function (uniformly recursive in p) defined by J/ (0) = P (e (0)) 3
136
[Ch. 11
CONSTRUCTIVE ORDER TYPES
where m,> >m,. Since every integer >O has a unique dyadic expansion, $ is well-defmed. Ifx=C(lxo ...x,1) let
...
t(x) =pz(0
Iz s r
+ 1 &(Vi)(z Ii 5 r = > x i = $ ( r - i)]
and be undefined for all other x and, further, writing t for t (x) since no confusion will arise, let if t = O ,
if t = r + l , be undefined for x not of the above form. Finally, let
D = {<x, u>:(x, u> fdX & (VZ) .(
4 4z)
Y
* +)))
,
then the reader will readily verify that q:W*DN X, where 4 (An, 4)=P (sn(4). LEMMA. 2'= X* W x=X or X= W. PROOF. If 2'= X,then X= WD for some D by the previous lemma. Hence 2wD= WD =X.But 2wD=(2w)D= WDby theorem 7.4.2 and hence W D = WD.Now D >0, since X> 0, hence for some E, D = 1+E and
11.2.3
W D =
p + e = W.WE= WD.
By theorem 8.3.5 it follows that W E= D = 1 + E.
If E=O; then X= W; otherwise E>O and by corollary 10.1.7 we have 1+ E = W E = 1 + W E .
Ch. 111
E-NUMBERS
137
Therefore, E= W E =D and
Thus (ii)*(i.ii). Using a related sort of argument to that used to prove lemma 11.2.2 we prove (iii)*(iv). The essential point is that any number not representing an &-number(in the ordering X) is expressible in Cantor normal form in terms of numbers representing (strictly) smaller ordinals. 11.2.4 LEMMA. If W x = X , then X=E(A) for some A. PROOF.Suppose W x = X and X E X , then there exists a recursive isomorphism p such that
p : x -wx.
We may assume min(X)=O. Let
and let q be the partial recursive function defined by
and undefined otherwise.
(5)
Then it is easily proved that 6 q z C'E(A) (by transfinite induction) and the q maps C'E(A) into C'W'. q is one-one. We prove this by induction on the maximum number n of applications of the cases (1)-(4) above in the computation of the q(xi)necessary to compute q(x). If n =0, then only (1x3)arise. The only difficulty is if cases (2), (3) conflict. But e(!)
138
CONSTRUCTIVE ORDER TYPES
[a. 11
represents the ordinal 1 in Wx so the right hand side of (2) represents w. On the other hand e
(3
,if it is in C'E(A), represents a fixed point of the
isomorphism induced between the ordinals by p by the condition in (3) and therefore represents an €-number in Wx, so there is in fact no contlict of requirements. Now suppose q(x) = q(y) = e
(m, ... m, 1'
then the ai, mi and r are uniquely determined and each ai = q ( x i )for some xi by condition (5). Moreover, the xi are uniquely determined by the induction hypothesis. Thus we see that r, the xi and the miare all uniquely determined and since e is one-one we conclude x = y . Finally, we leave the reader to prove by transfinite induction that 4:E(A)
= Wx,
whence E (A) N X. Putting A =COT (A) yields the required result. Now we show (iv) =-(iii).
LEMMA.W E ( A=)E (A). PROOF. Suppose AEA, then let f be the partial recursive function defined by
11.2.5
and undefined otherwise.
Ch. 111
E-NUMBERS
139
Then it is readily verified as in the preceding proof that
f:E(A)
1:
WE(A),
from which the lemma follows. Finally we show that (iii)
(ii)&(i). This requires a number of lemmata.
LEMMA. BA=A* (AB)W=AW. PROOF. ([email protected]" = (AB)2+W = (AB) (AB) (AB)W = A ( B A ) B(AB)" = A (AB) by the hypothesis, = A(AB)W. By lemma 7.4.1, it follows that A" divides (AB)" and, since lA"l= I(AB)wI (by e.g. BACHMANN, 1955, p. 57) it followsfrom theorem 8.3.3.(iii) that A" = (AB)". 11.2.6
11.2.7 LEMMA. B+A=A&A= WD+n*(A+B)W=AW. PROOF.We leave the trivial case D = O to the reader. We first observe that if B+ A = WD then (A
+ B)AW = ( A + B ) (WD)" = ( ( A + B ) W ) D(WD)" by theorem 8.2.2 with A W for C ,
= ( A W ) D(WD)W = ( W D W ) D(WD)"
= ( W D ) 2 + w= (WD)" = A".
By lemma 7.4.1 it follows that (A+B)" divides AW and hence (by e.g. BACHMANN, 1955, p. 57 and theorem 8.3.3.(iii)) that A"
=(A
+
Now suppose A = WD + n where n >0, then
+ ( A + B)"
(A
Hence
= (WD
+ n + B)"
= (WD
+ B')W
where B'=n+B.
140
[Ch. 11
CONSTRUCTIVE ORDER TYPES
If B+A =A, then
B+WD+n=WD+n and hence by theorem 5.1.6, B + WD= WD. Therefore B'+ W D = n + B + W D = n + W D = WD by lemma 7.4.1 (since we are assuming DPO). Putting A'= WD we immediately obtain from the first part of the proof (A' i.e.
+ B')w = A f w ,
(WD + n
i.e.
+ B)W = (WD)W,
( A + B)" = (W D ) W .
Finally n+ WD= WD and hence putting B=n, A = WD in the first part of the proof we get (WD + n)" = (WD)" . (7) Combining (6) and (7) gives the desired result.
11.2.8 LEMMA.Wx=X=+X is a principal number for addition and multiplication. PROOF. This follows at once from corollaries 10.1.7 and 10.2.8. LEMMA.W x=X& 1 c Y <X* Yx=X. PROOF. By theorem 9.2.8 the hypothesis implies
11.2.9
y = ~ ~ 1 +..-+ . n WX*-n, ~ where r 2 1, X> X,> ...>X,and the n, are finite, non-zero. I f r = l , X,=O, then Y = n , > l , hence yx = nT = n"x - "*"X = W"X = 1 -n1
x,
since W x is a principal number for multiplication and W is a principal number for exponentiation by theorem 7.4.2. In any other case we have yx = y w x = (Y")WX = ((W"1.n , ) W ) W X
=(<wx,)w )wx = WXlWWX = WXlWX = wwx =X.
by lemma 11.2.7, by lemma 11.2.6, by lemma 11.2.8,
Ch. 111
141
E-NUMBERS
Putting Y=2 we get (iii)*(ii) and the stated version of the last lemma gives (iii)*(i), so the proof of the theorem is complete. 11.2.10 COROLLARY. The collection Z(ex p ) of all principal numbers for exponentiation is strictly &,-unique. PROOF. By theorem 11.2.1 every principal number Wis of the form E (A)for some co-ordinald. By theorem 11.1.3if IE (A)I (X)i
* (x),)]
.
For any x (be it a sequence number or not) the predicate E(y, x ) : “ y extends x &Seq (y)” defined by E(y, x)*Seq(y)
& Zh(x) 5 Zh(y) & ( V i ) ( i < Zh(x)=>(~)~= (y),)
is (primitive) recursive. Hence there exists a recursive function t (n, x ) such that t(n, x ) > n a n d w , ( , , , = { y : y € o , & E ( y , x ) } . Main construction. We define a sequence of integers ai such that { (ao,...,a,, 1)},“= has limit A. (We add the 1at the end simply in order to avoid violating the definition of sequence number.) Stage 1. a,=O. Stage n+ 1. Suppose a,, ..., a , , , - , have been determined and t(n, (a09 ... a m ” - * ))=y (wherey>n). a) If T (y) then enumerate a,,until three distinct elements u< u < w 9
Ch. 141
171
QUASI-PINITENESS
+
have been obtained. Let m 1= max (lh (u), lh (u), lh ( w ) ) , (m is always defined since a, has been determined.) If u=(a,, ..., a,,- I , ..., ak) then a,, ..., ak are thereby. determined and if k 1<m then set
+
ak+,
= - a * =
a, = 0 and
m,, = m .
b) If 1 T(y) then set mn=m,,-, thereby determining no new ai. 14.2.5 LEMMA.For each n, a,, is eventually determined and no a,, is ever changed once determined. PROOF. a, is determined. If a,, ..., a,,,-, have been determined then since every set is repeated inllnitely many times in the enumeration oi we can always find 3 extensions of any given sequence number. Thus a,, ..., a,,,- has a proper extension. The first part of the lemma now follows by induction and the second assertion is trivial.
Letus defineA,={x:xA} to be thelowerandupper Dedekind sections determined by A. 14.2.6
LEMMA.A,, A,, are not r.e.
PROOF. If A, is r.e. then A,=o,, for some n and T(n) holds since A,
is infinite. Thus o,, contains three distinct sequence numbers u< u < w. Extend the sequences by 0’s to u’, u’, w’ so that lh (u’)= Ih (v), lh (w))= m + 1 and abbreviate m,,-l by r ; then U’=(ao,...tar, b r + 1 , * * * ,bm), u‘ = (a,, ...,a,, a,+,, ..., a,), w’ = (a,, ..., a,, c,+,, ..., c,),
say.
Now A lies strictly within the open interval (of real numbers) (u, u) since u’, u’, w’ all differ and (a,, ..., a,,) for n>m extends u’. Hence w$A, which is a contradiction. Similarly A,, is not r.e. We note that since A lies in the intersection of all the intervals (u, w ) the set {(a,, ..., a,, 1)},“=, has precisely one limit point. Now the procedure for constructing A is clearly recursive in El hence by lemma 14.2.3 A is A 2 , thus the predicate A(a):“A extends a” defined by (3n) (n 2 Ih(a) & E(O. PROOF. By theorem 14.2.2there exists a recursive quasi-finite C.O.T. A, say, of type o+o*.By theorem 14.2.7 A - n is a recursive quasi-finite C.O.T. for every n and by corollary 2.4.12, A.n=B-n&n>O implies A=B so it suffices to prove there exist KO recursive quasi-finite C.0.T.s of type w + a*. Let A be a r.e. quasi-finite linear ordering of type o+o*and let B be 14.2.8
Ch. 141
173
QUASI-FINITENESS
A [B where B is any infinite r.e. proper subset of C'A. Since C'A is infinite such a B exists. B is clearly a r.e. linear ordering. But if B contained a recursive ascending or descending chain then so would A hence 6 is r.e. and quasi-finite. But B A by theorem 14.1.4. Clearly, we can then repeat the process to obtain K O such B, such that A 3 B i 3 Bj for i< j , but A+ Bi+ B, if iij . Taking C.0.T.s yields the required result.
+
14.2.9
THEOREM. If A is quasi-finite then A
=B
+A +
C=O*B
=C=O.
PROOF. By theorem 3.2.7, A = B + A + C implies C=O. By theorems 3.2.7 and 2.4.2, A = B + A implies B=O.
14.3 We defined quasi-finite C.0.T.s as those C.0.T.s A such that both A and A* are quords, so most of our results for I obtained in previous chapters for quords also apply to I* for quasi-finite C.0.T.s with the obvious exceptions caused by the failure of closure conditions, etc. is We shall omit detailed mention of the results. But we note that I* reflexive, anti-symmetric and transitive on quasi-finite C.0.T.s and is a tree ordering. It is not a linear ordering or a partial well-ordering. The former follows from theorem 14.2.8 for let A be a quasi-finite C.O.T. of type ( 0 + 0 * ) * 2 then if AEA there exists B c A of type o * + w . Clearly A and COT(B) are incomparable under I* since they have final segments with incomparable (classical) order types. I* is not a partial well-ordering for if A is a quasi-finite C.O.T. as given by theorem 14.2.8, then A is expressible in the form n B,, for each n where B,, is quasi-finite and
+
B, (Vj) (jls*(Vz) (ZETsi n6pj*t* pj (2))) (d) t B { pr ( Z r ) :Y I S } , where Zk = V,,,{(Pk(W)iS defined & p k ( w ) # { ( p j ( Z j ) : j < k } U { X j l : j < k & 1E2'} 7
U{xkt:
1l,s=t. (Thereasonwehave to take y + 1, z+ 1 rather than y, z is that the minimum element of EF
188
CONSTRUCTIVE ORDER TYPES
[Ch. 16
when E, F are finite exists and is 0 and we do not have OEB', OEC' in Cons (5)). As before we obtain (l),(2), (3) and since p is order preserving everywhere it is defined, we also have
and
biu and ci, > c c . . . > ~ c i u
bi,
for all possible bracket symbols obtainable from A*, B*, C*. If Cons (5) is finite then p is an order isomorphism between finite sets (here we do not need to bother about adding 0 as minimum element) and hence, by lemma 16.1.1, unique. Finally we define
if Cons(xeB) is finite, undefined otherwise. The rest of the proof now follows analogously to that of theorems 16.1.2 et seq. and we are through.
16.4 This section is devoted to the proof of a cancellation metatheorem for losols. The idea of the proof is just like the ones for previous cancellation theorems, namely, given P(X)G P(Y) we look at all possible elements (effectively)"generated" from a single element X E C'X and then, since C'X is isolated this set is finite so we can use the corresponding cancellation theorem for finite sets (or numbers) and thence obtain a recursive isomorphism from X into Y. Notation: u;(xl,
...)x")=xi.
16.4.1 LEMMA. If p is a (number-theoretic) non-constant one-place function obtained by finite compositions of the functions
u;, x + y, x - y ,(1 + x)'
-1
and (finite) parameters, then p(x) I p(y)
if, and only if, x 5 Y
PROOF. We proceed by induction on the complexity of the function p .
Ch. 161
189
ARITHMETIC LAWS FOR LOSOLS
Clearly Ul has the required property when all arguments except the ith are held constant. All the other functions listed are strictly monotone increasing in each argument or constant so they satisfy the condition. Now suppose h (x) = Uf(fi (x, i i ) , ...)f n ( x , i n ) ) 7
where the f i are functions obtained by composition from the given functions and the di are sequences ail,...,aikrof finite parameters. Then h (x)=L( x , di) so this case reduces to a simpler one. Finally suppose h ( X I = 9 (fl(x, i l ) , f z (x, 4))3 where at most one offl, f z is constant and the rest ofS,, fz satisfy the condition. Then h ( x ) is strictly monotonic increasing since it is increasing in both arguments and thefi are increasing too. Hence x I y-h(x)
Ih(y).
We note that these are the only cases we have to consider since we do not need to distinguish between constant functions and parameters as far as their values are concerned. 16.4.2 THEOREM (CANCELLATION METATHEOREM FOR LOSOLS). If P is a non-constant one-place function (from losols to losols) obtained by finite compositions of the functions
ui”,x + Y,x - Y ,(1 + X)‘
- 1, x*
and (losol) parameters, then (i) P(X)G P(Y)if, and only if, XG Y and (ii) P(X)=P(Y)if, and only if, X= Y. PROOF. Suppose the losol parameters involved in P are Al, ..., A,. Then, since P is non-constant we must have certain of the Ai >O. We shall assume that in this case if A,EA,>O then min(A,)=O. We shall also assume that all the linear orderings we consider are E R but that min (X), min(Y)*O. We associate with P a set of canonical embeddings of X in P(X). P is a function from linear orderings to linear orderings inducing P . We shall assume without loss of generality that P is obtained by composition analogously to P where the basic functions correspond as
190
CONSTRUCllVE ORDER TYPES
[Ch. 16
below.
u; ,
U/ X+Y x.Y (1 + X)' - 1 X*
X +Y rn (X, Y) e(X, Y) r (XI
(see theorem 2.2.2), (see theorem 6.1.6), (see below), (see below).
Let R denote R[(Seq-(0)) then R, R'* are recursive dense linear orderings without first or last elements and so by theorem 1.2.4, for some r, r : R'*E R'. We take r to be fmd. This is the r of the table above. Since we are assuming min(X)+O we can take as representative of l + X , X+={a,EC'A,, & a , =k min A,,)}. ( K may be the empty bracket symbol.) Now define
II(A) = {(e(K), e(K')):K , K ' E E ( A )
B.2 LEMMA.If (A,:C) is a standard sequence ordering with C'C r.e. and C has a minimum element, co, say, then II(Ai: C) II AC;II(Ai:
C[(CC- {cO})),
204
[AP. B
A. 0. HAMILTON
PROOF. By our definition of standard sequence ordering every A i R ~ C E R). Let f be the function defined by
(and
'
(: ... ""-'>)if c, co each c i ~ C ' C , j(min(Aco), "'i:)) if c, co ... j (a", e
=
*"
u"-l
&
=i=
e(i:
& each ci E C'C , undefined if any ci 4 C'C or if the argument o f f is not of the above form.
I
It is easily verified that f is one-one and maps CII(Ai:C) onto C'A,;II(Ai: C [(CC- {co})), Since, by hypothesis C C is r.e. and Seq = C'R is r.e. it follows that 6f is r.e. Finally we leave the reader to check that f is order preserving on the whole of its domain. B.3 COROLLARY. With the hypotheses of lemma B.2, if D E C and min (D) = co then
II(Ai: D)
N
Aco.II(A,:D [(C'D
- (~0))).
PROOF. The same function f as in the proof of the lemma is the required recursive isomorphism. Notation: If C has a first element, we shall denote C[(C'C-{min(C)}) by C'. If C has an initial segment of type n (finite) and n > 1, then C(") denotes (C'"- I))'.
B.4 COROLLARY. Let C be a well-ordering of type o (cR) with r.e. field, and let D be a sub-ordering of C with an initial segment do,..., d,,-l (i.e. the first n elements are d o . . . in that order). Then
n ( A i : D)NA,j;**-.Ad,- ;II(Ai: D'"'). PROOF.If n = 1 , let C , = C [ ( X : X E C C & ( ~x)EC}. ~, Since do can have at most a finite number of predecessors in C, C, has an r.e. field and we can apply corollary B.3 to get
II(Ai: D)
N
Ad;n(Ai: D').
Now assume n > 1. Suppose, as induction hypothesis, that
II(Ai: D)
N
Ado*..*.Ad,-2.n(Ai: D("-')>.
AP. BI
INFINITE PRODUCTS
205
We have to show that
But this follows immediately from the first part of the proof. So we conclude n ( A i : D) N A,,.*.*.A,,-,*II(Ai: D'"'). We now define a collection 42 of sub-orderings U of W' as follows: By theorem B.5 below, there exists a collection of 2'O isolated sets none of which is mapped into any other by a finite-to-one partial recursive function. We can suppose that none of these sets contains 0, 1 or 2. Now order each set by magnitude, and we get 2'O well-orderings of type o.For each such well-ordering U' let U = {((rn), (n)) :(rn, n ) ~ u ' } . Then the collection 42 of all such U has 2'O members, each of which is of type o,r W but not recursively isomorphic to W and moreover no two members of 42 are recursively isomorphic and no finite-to-onepartial recursive function maps the field of one into the field of any other. Every U E 42 satisfies the hypotheses of corollary B.4 which apply to D (with W' as the well-ordering C). B.5 THEOREM. There exists a collection d = { A i : i E I }of sets of (nonnegative) integers with the following properties: (i) the cardinal of I is 2'O, (ii) for all i , j ~ with I i + j there is no finite-to-one partial recursive function mapping Ai into A j , (iii) each Ai ( ~ E I is ) isolated. PROOF. Let cpo, cpl, ... be an enumeration (non-effective) of all finiteto-one partial recursive functions with infinite range. "20/
We construct a tree
which branches into two at each node, each node being an integer. The sets Ai will be the branches of the completed tree.
206
LAP. B
A. G. HAMILTON
Stage 0. Let ro be any integer not equal to cpo(zo)where zo = v, { q o ( w ) is defined).
We describe stage s for s 2 1. Stage s. We define successively Yo, Y1, ..., Yz'-'. Define 2' for 0 5 i l 2 ' - 1 as follows : Let Tsi=the set of all elements of the tree which have been defined before xsi, B,,=the set of all elements defined before Y i which lie on the same branch as x", TL=the set of all elements of the tree defined when stage s- 1 has been completed. Choose 9' to be some (say the least) integer t satisfying (a) t#TSi, (b) either t$ U d q j or t E U 6 q j and (Vj'jrs) (t&pj=.qj(t)$Tsi), jss
jss
*
(c) ( V j s8 ) ( V Z ) (zeTsi nh j * q j ( z ) t ) , (d) t $ { q j ( z j ) : O S j s s +l), where .zj=vw(qj(w) is defined and (Pj(w)#(qk(zk)
Ik<j) uT(i3.
At every stage we have infinitely many numbers to choose from; each of conditions (a)-(d) disqualifies only a finite number of integers since each j is finite-to-one with inlinite range. Thus such a t always exists. This describes the construction of the tree. The branches of this tree are the sets Ai. (i) The cardinal of I is 2N0since all the branches are distinct, (ii) Suppose q k : A i s A where j i i j . Then Ai*Aj by the construction, and we may find x€Ai-Aj such that x appears in the construction of the tree after the kth stage. Then y = q k ( x ) ~ Aandy+x. j Ify appears in the tree before x, then (b) is violated when x is put in the tree. If y appears after x then (c) is violated when y is put in. Thus we have a contradiction so condition (ii) of the theorem is satisfied. (iii) Suppose Ai is not isolated, then Ai contains an infinite r.e. set, say p ( p k (it must be the range of one of the 9;s). At stage k we kept Pk ( z k ) out of every branch of the tree, and at dl later stages. But z k was chosen so that qk(zk)did not occur in the tree before stage k. So qk(zk)$AI which is a contradiction. Thus condition (iii) of the theorem is met and the theorem is proved. We now write ui for the ith element of U in the obvious way.
AP.BI
207
INFINITE PRODUCTS
B.6 LEMMA.If UEQ, {Pi},,, is a strictly increasing sequence ofprincipal numbers for multiplication and PiEPi for all i, then COTJI( Pi: U) is a principal number for multiplication. PROOF. We have to show that if B is recursively isomorphic to an initial segment of II( Pi: U) then B*II(Pi: U)
N
II(Pi: U ) .
Suppose
B N B' < II(Pi: U> and suppose
U = {(ui, uj): i I j } . We must have
B1 P,, for i < j - 1. Now, using corollary B.4 and properties of principal numbers we have:
B*II(Pi: U ) N B'.II(P,: U) N B' .P , .P,, * * P-, ,* II (Pi :U"') N B'*P,,-,*II(Pi: U"') N Pu,-,.n(Pi:U"') N p,,..... PUj-;II(Pi: U"') N II(P,: U). Hence COTII( Pi: U ) is a principal number for multiplication.
B.7 LEMMA.If B N WAfor some A, then the function h a 2 is recursively representable on B. PROOF. If B=WAthen C'B consists of elements of the form
and the ni (01i s m ) are non-zero integers. The function f defined on pe as follows recursively represents the
208
[AP.B
A. G . HAMILTON
function Iaa2 on B: fe
("no ... n,
am)
=
(2n0n, .*.... n,
am)
a0 '1
'
i.e.
If
(x)lB= (X(B.2 whenever XEC'B
and f(x)$C'B
if x$C'B.
This function f is obviously one-one and partial recursive mapping C'B into C'B. That it represents laa2 on B can be seen by mapping the set i x : (e
(an: ::::),
x)
EB
'"'))
(
& (x, e 2n0n, a '.'.*. n,
in a one-one order preserving way onto the set
( ('
y : y , e no *... ** n,
E
EB
1,
1
B ,
as follows
Now if g : B ' z B=WAthen g-'fg
recursively represents lam2 on B'.
B.8 LEMMA.If [email protected] is a strictly increasing sequence of principal numbers for multiplication, then Pi€Pi can be chosen so that the function h a 2 is not recursively representable on II( Pi: U). PROOF. Let U = {(ui,u j ) : i s j } .Since all the Piare principal numbers, we must have Pi > 3 for all i, so we can choose the P,EP,with the following properties: min(Pi) = 0, min(PI) = (1) = 2, rnin(P1')) = u, . That we can choose orderings thus and still have them included in R is n easily seen (e.g. if QiePi, take ui+2Qi and replace the first three elements by 0,2, ui. The order must remain correct since ui >2 for all i.) With this choice of { Pi)i<w,consider
AP. Bl
209
INFINITE PRODUCTS
Suppose a is the ordinal represented by e Then the element representing a2 is e e(
i)
can be mapped exactly twice in a one-one order preserving way into
the set of predecessors of e (co a,
(3
by the identity function and the function
... cn) ... an
--t
(j c o ...... 2a,
Cn)
an
Now if 1aa2 were recursively representable on lT(P,:U), function defined on numbers of the form e
(i),
then the
taking e(i) to e
(L)
would be partial recursive. However, this would imply that Xui was a (partial) recursive one-one function. But U is not recursively isomorphic to W, so liuiis not a recursive function. Therefore h a 2 is not recursively representable on ll( Pi: U).
B.9 COROLLARY. COTII( Pi:U) is a principal number for multiplication not of the form W A . PROOF. By lemma B.6, COTII( Pi: U) is a principal number for multiplication. Suppose it were of the form W A ,then
n(Pi:U)=WA
forsomeA.
(*I
By lemma B.8 1aa2 is not recursively representable on 11(Pi: U). By lemma B.7, 1aa2 is recursively representable on WA.This contradicts (*) and the corollary is proved.
+ COT II ( Pi : U) + COT II (Pi:
B.10 LEMMA. If U, V E 42 and U V then
V).
PROOF. Suppose the contrary, then there is a partial recursive one-one function f such that f:II(P,: U) N II(P,:V). Now
fe
(9) ( ... ") = e co'*' a, a,
for some c,, ..., c , ~ c ' v and aiECLP,,
210
[AP. B
A. G . HAMILTON
so and therefore
is a finite-to-one partial recursive function mapping C'U into C'V. For suppose there were a v, E C'V which was the image of infinitely many u~EC'U.It would then follow that all of II( P,: U> is mapped onto a proper initial segment of II( P i : V) by f, namely that initial segment determined by e
pi1)
(since U has type w and the set of ui mapped
to v, is cofinal with U). This is a contradiction, so AxZ((fe ($,)
is
finite-to-one. That it is partial recursive is obvious. But the existence of such a function contradicts the definition of 4. Hence there is no such f and therefore
COTII(P,:U>+COTII(Pi:V). B.ll THEOREM. There exist 2" principal numbers for multiplication of classical ordinal ow"which are not of the form WA. PROOF. By corollary B.9 COTII( P i : U) is a principal number for multiplication not of the form W A ,for each UE 4. By lemma B.10 all the co-ordinals COT n
....- p ~ m - p ~ + P + l - . . -where - p ~ + ,a,,,,
b, =k 0.
We make a lot of use of Kleene's indefinite description operator v, which from a partial recursive function f yields a partial recursive function 9 such that ( 3 Y ) ( f ( Y ) = 0)=. 9 (V,(f(Y) = 0)) = 0 *
214
NOTATION
We define x ~ asy usual by x-y=x-y
=O but we also define x
7
if x k y , otherwise ,
y = pz { y + z
= x}.
So x - y is a partial function in general.
Other unexplained notations may be found in KLEENE (1952) especially p. 538.
REFERENCES
P. H. G. ACZEL,1966,D. Phil. Thesis, Oxford. 1966a,Paths in Kleene’s 0, Archiv Math. Logik Grundlagenforschung 10, 8-12. - & J. N. CROSSLEY, 1966,Constructive Order Types, 111, Archiv Math. Logik Grundlagenforschung 9, 112-1 16. H. BACHMANN, 1955, Transfinite Zahlen (Berlin). G. CANTOR,1915, Contributions to the Founding of the Theory of Transfinite Numbers, (Dover Reprint). 1936,Formal Definitions in the Theory of Ordinal Numbers, A. CHURCH& S. C. KLEENE, Fundamenta Mathematicae 28, 11-21. J. N.CROSSLEY, 1963,D. Phil. Thesis, Oxford. -, 1965, Constructive Order Types, I , in Formal Systems and Recursive Functions, Eds J. N. Crossley&M. A. E. Dummett (Amsterdam) 189-264. -, 1966, Constructive Order Types, 11, JSL 31, 525-538. - &R. J. PARIKH, 1963, On Isomorphisms of Recursive Wefl-orderings (Abstract), JSL 28, 308171. - &K. SCHUTTE,1966,Non-uniquenessat w2 in Kleene’sO, Archiv Math. Logik Grundlagenforschung 9,95-101. J. C. E. DEKKER& J. MYHILL,1960, Recursive Equivalence Types, Un. of California Publications in Mathematics, n.s. 3, 67-214. A. EHRENFEUCHT, 1957, Applications of games to some problems of mathematical logic, Bull. Acad. Polon. Sci. 5 pp. 35-37. E. ELLENTUCK, 1963,Solution of aproblem of R. Friedberg, Mathematische Zeitschrift, 82, 101-103. S. FEFERMAN, 1968,Systems of Predicative Analysis, II; JSL 33, 193-220. R. FRIEDBERG, 1961,The Uniqueness of Finite Division for Recursive Equivalence Types, Math. Zeitschrift 75,3-7. R. 0.GANDY, 1960,Proofof Mostowski’s Conjecture,Bull. Polon. Acad. Sci. 8,571-575. A. G. HAMILTON, 1968, An unsolvedproblem in the theory of constructive order types, JSL. 33, 565-567 S. C. KLEENE,1952, Introduction to Metamathematics (Amsterdam). -, 1955, On the Forms of Predicates in the Theory of Constructive Ordinals (Second Paper), Am. J. Math. 77,405-428.
216
REFERENW
G. KREISEL,1960, Non-uniqueness Results for Transfinite Progressions, Bull. Polon. Acad. Sci. 8,287-290. J. MCCARTHY, 1956, The Inversion of Functions defined by Turing Machines, Automata Studies, Annals of Mathematics Study no. 34, (Princeton) 177-181. A. NERODE,1961, Extensions to Isols, Annals of Mathematics 73, 362403. R. J. PARIKH,1962, Some Generalisations of the Notion of Well-ordering (Abstract), Notices Am. Math. SOC.9, 412. -, 1966, Some Generalisaiionsof the Notion of Weil-ordering,Zeitschrift Math. Logik Grundlagen der Mathematik 12,333-340. H. G. RICE,1956, Recursive and recursively enumerable orders, Transactions Am. Math. SOC. 83,277-300. K.ScHUme, 1965, Predicattve Well-Orderings,in Formal Systems and Recursive Functions. Eds J. N. Crossley and M.A. E. Dummett (Amsterdam), 280-303. W.SIERPINSKI, 1948, Sur la Division des Types Ordinaux, Fundamenta Mathematicae, 35,l-12. -, 1958, Cardinal and Ordinal Numbers, (Warsaw). R. I. SOARE, 1969, Constructive order types on cuts (to appear in JSL). C. SPECTOR, 1955, Recursive Well-Orderings,JSL 20, 151-163. A. TARSKI,1949, Cardinal Algebras, (New York). -, 1956, Ordinal Algebras, (Amsterdam). J. S. ULLIAN,1960, Splinters of Recursive Functions, JSL 25, 33-38. 0. VEBLEN,1908, Continuous Increasing Functions of Finite and Transfinite Ordinals, Transactions Am. Math. SOC.9, 280-292. A. N. WHITEHEAD & B. RUSSELL, 1927, Principia Mathematica, Vol. I1 (Cambridge).
NOTES
INTRODUCTION 01 This is closely related to the results of CROSSLEY AND PARIKH (1963). 02 The term “recursive isotonism” which has been used instead of “recursive iso(1963, 1965), ACZELAND CROSSLEY (1966) and some morphism’’ in CROSSLEY other places has a slightly different meaning. 03 This will be exploited more in work by NERODE and the author which is in preparation. 04 We abbreviate “recursively enumerable” by “r.e.”.
CHAPTER 1
11 The author has discovered (July 1966) that this theorem was essentially proved by NCE (1956, theorem 20) and an essentially identical one by SPECTOR (1955). 12 There will be no confusion between sequence numbers of sequences of two elements and ordered pairs since the context will always make clear which is intended. is defined on p. 213. CHAPTER
2
21 Note that this notation will causeno confusion for if A is set-theoreticallyincluded inBthenA=B. 22 We use the word “step” here in the sense of a whole phase in the calculation rather than moving just one square on the Turing machine tape. CHAPTER 3 31 The quantifier (Vf) ranges over all one-place functions and the quantifier (VB) ranges over all sets of natural numbers. 32 This use of the word “splinter” is derived but differs from that in ULLIAN (1960).
218
NOrnS
CHAPTER 4 41 Such exists by theorem 3.1.5.
CHAPTER 5 51 Although these are not quite the same orderings as in CROSSLEY (1965), the
co-ordinals are the same. 52 Since we get most of our counterexamples from V. 53 This simple group-theoretic way of presenting the proof is due to Alex Rosenberg.
CHAPTER 6 61 By “minimal” we mean minimal with respect to domain and range. 62 This argument is basically due to TARSKI(1956). 63 This argument and those to the end of this chapter are due to TARSKI (1956)
though the proofs of the supporting theorems are very different from his. CHAPTER
7
71 Since we are assuming min A =0 we shall never misidentify two sequences since no sequence can end in 0. 72 This is a strengthened version of theorem VIII.2.2 of CROSSLEY (1965).
CHAPTER 8 81 This is inspired by TARSKI(1956). Compare also theorem 2.4.11. 82 We are here using an extension procedure similar to that in the proof of theorem 8.3.5. 83 We write e(A, B) for { e ( A ) : AE (A, B)}. 84 Recall ( x ) ~ =exponent of 2 in prime factorization of x. CHAPTER 9
91 In fact we show that many co-ordinals may be expressed as polynomials in W
with large exponents. 92 This theorem was first conjectured by A. L. Tritter. 93 We are assuming min(A) =0 as usual. CHAPTER 10 101 This definition is adapted from KREISEL (1960). 102 The problem of whether there exist principal numbers for multiplication not of the form W Ais solved affumatively in the appendix B by A. Hamilton.
CHAPTER11 111 The results in this chapter were obtained by P. H. G. Aczel and the author and
NOTES
219
(1966). The name “Eappeared in their original form in ACZELand CROSSLEY number” is intended simply to convey that these co-ordinals are closely related to (classical) .+numbers. It should not be confused with the E(I), etc. of VEBLEN (1908). 112 ea =PO (09 = / 3 & / b cy for y< a} or, equivalently, the a-th (classical) principal number for exponentiation greater than w. 113 Here XI 2 E . . , 2 is an abbreviation for (XS, X I > E
E&.
..&<xr, X~-I>E E .
CHAPTER 12 121 7 i s defined on p. 14. CHAPTER 13 131 The reader who is familiar with R.E.T.s and isols is advised to omit this chapter. 132 There will be no confusion although we use the same type founts for (e.g.) R.E.T.s and C.0.T.s since the context will make clear which is intended (generally, R.E.T.s in this chapter). 133 Here again we are dealing with unordered sets so N is not ambiguous.
CHAPTER 14 141 The construction given here is due to C. G. Jockusch. APPENDIX A1 The question whether the theorem holds with the weak sense of 5 is open. REFERENCES 171 See the correction in the author’s abstract JSL 31,292-3.
INDEX OF SYMBOLS
8,
0 W1
8L V
1
5 5 6 211 211 211 211 211 211 211 211 211 211 211 211 211 211 21I 211 212 212 212 212 212
212 212 212 212 190,212 26, 32, 189 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212
221
INDEX OF SYMBOLS
212 212 213 213 213 213 213 213 214 214 21 22 22 22 22 23 23 25 26 26 26 26 27 29 30 30 31 32 32 32 34 34 34 38 38 38 29 43
46 46 46 46 212 52 54
54 54 54 54 56 56 56 56
(b...'bn),(A, B) ao...an
56 56 60 62 62 63 75 75 76 76 76 76 77 79 80 82 93 93 97 97 103 103
222
As,B A = (Ai :C) S.O. s.W.O. CST(A)
z %4 0, (for R.E.T.s) R (R.E.T.) A + B (for R.E.T.s) I (for R.E.T.s) E for C.0.T.s N
on
xi, ni, Ai T(4 E(Y, x) A, A, 9 I ,I *, redefined
Seq'
INDEX OF SYMBOLS
133 142 1 42 142 143 144 144 145 145 159 160 161 162 168 170 170 170 170 171 171 176 177 178
INDEX OF TERMS
Aczel, P. H. G., 5, 6, 12, 16, 17, 22, 131, 142,153, 175, 194 addition of C.0.T.s 12, 34 - of C.S.T.s 146 - of R.E.T.s 161 antisymmetry 21, 68 ascending chain 167 Bachmann, H. 11,92, 130 bound, upper 16, 150 - -, existence of 152 bracket symbol 75 cancellation laws for C.0.T.s 92 et seq, 194 et seq - - for isols 165, 166 - - for losols 182 et seq - - metatheorem for losols 189 Cantor, G. 12, 24 Cantor normal form 15, 110 et seq, 120 cardinality of Q, etc. 47 Church, A. 11, 122 class 211 classical results 11 cofinal212 collection 212 commuting co-ordinals 57 constructive means 6 - order type 12, 23 - - _ , standard 29 - sequence type, 16, 143
- - -, standard 145 converse 37, 167 co-ordinals 13, 46 -, commuting 57 -, incomparable 55 -, natural 54 -, recursive 46 -, sequence 143 counterexample, basic 54 Dedekind section 169 Dekker, J. C. E. 5, 12, 16, 23, 159 et seq, 177, 178 descending chain 41 directed refinement theorem, 36 - - -, generalized 148 disjoint sets 212 -, strictly 30 dots 21 1 dual 38, 195 Ehrenfeucht, A. 16 Ellentuck, E. 166 E-number 16, 131 et seq. exponentiation of C.0.T.s 15, 75 - of R.E.T.s 165 Feferman, S . 5 field of a relation 21 final segment 38
224
INDEX OF TERMS
function, partial 212 -, total 212 -, domain of 212 -, range of 212 Gandy, R.0. 13. 15 generates 29, 195 Hamilton, A. G. 17, 18, 176, 178,203 immune set 213 infinite product 203 et seq - sum 144 et seq initial segment 38 isol, 16, 164 et seq isolated set 160 - linear ordering 176 isomorphic 21 isomorphism, classical 15 -, of sequence orderings 142 -, recursive 11, 22 -, recursive, pair 143 - with classical ordinals 129, 141 Jockusch, C. G. 17, 169 k-attained 79 Kleene, S. C. 5, 25, 26, 43, 78, 122, 143, 170, 179,211 Kreisel, G. 5, 6, 122, 167 A-notation 21 1 limit number, 52 Lindenbaum, A. 13,39 losol 16, 175 et seq McCarthy, J. 23 minimum element 25 Morley, M. 12, 13, 39, 40, 94 multiplication of C.0.T.s 14, 60 - of R.E.T.s 162 Myhill, J. 5, 7, 12, 16, 17, 23, 159 et seq, 175, 177, 178 natural well-ordering 5, 54 - co-ordinal 54 Nerode, A. 7, 17, 29, 166, 167, 175, 176, 178, 193 notation 21 1 -, unexplained 214
order embedding 194 - type 22 _ _ of quords 13 ordering, dense 25 -, linear 21 -, natural well- 5 -, partial 21 -, partial well- 49 -, rooted tree 49 -, sequence 142 -, sequence well- 142 -, tree 49 -, well- 41 ordinal algebra 5 -, classical 46 -, first non-recursive 6 - number 11 - recursive 11 - sum29 Parikh, R. J. 5, 6, 13, 15, 17, 42, 43, 78, 122, 167 partial ordering 21 - recursive functional for addition 34 _ - _ _ multiplication 62 - - - - exponentiation 77 - _ _ _ infinite sums 144 path 16, 56 -, long 153 pathological features 12, 28, 51 predecessor, weak 177 principal number 15 - _ for addition 92 _ _ _ multiplication 97, 203 et seq - - _ exponentiation 103, 135 quasi-finite, 17, 167 -well-ordering 13,41 _ _ -, recursive 42 quord, 13,43 -, recursive 43 -
recursive equivalence 159 - equivalence type 16, 159 et seq - isomorphism 11, 22 - isomorphism pair 143 - permutation 23 - sequence type 16 refinement theorem, directed 36
INDEX OF TERMS
- -,for R.E.T.s 161
- -,
generalized directed 148 relation 21, 212 -, field of 21 -, recursive 22 -, r.e. 22 -, reflexive 21 -, restriction of 212 Rice, H. G. 12, 25 Rosenberg, A. 58 Russell, B. 30 Schiitte, K. 6, 7 segment, closed 56 -, final 38 -, initial 38 -, open 56 separable 30 separation lemma 12, 35 - _ , generalized 148 sequence co-ordinal 143 - number 26 - well-ordering 142 sets 211 -, disjoint 212 -, empty 212 -, finite 14 -, immune 160, 213 -, isolated 160 Shepherdson, J. C. 5 Sierpinski, W. 11, 37, 39, 44, 57, 60,92, 98, 102, 194 Soare, R. I. 23
225
Spector, C. 13,43 splinter 44 standard C.O.T. 29 - C.S.T. 145 strictly disjoint 30 - a-unique 122 successor 53 - number 52 Tarski, A. 5, 13, 14, 15, 39, 68, 72, 94, 102, 147, 149 Tennenbaum, S. 17, 169 transitivity 21 tree ordering 49 - property 14 trichotomy law 21 Tritter. A. L. 111 Ullian, J. S. 44 unique factorization 130 -, a- 122 -,E W , co-ordinals 141 -, am-, co-ordinals 125 -, wWm-,co-ordinals 128 upper bounds 16,150 Veblen, 0. 5, 131 well-ordering 41
-. natural 5
-, partial 49 -, sequence 142 Whitehead, A. N. 30