Determinacy and Prewellorderings of the Continuum

DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM YIANNIS N. MOSCHOVAKISt University of California, Department of Mathematics, Los Angeles, California, U.S.A.

Let A be a class of subsets of R x R , where R = = the set of number theoretic functions. An ordinal 5 is realized in A if there is a relation in A which is a prewellordering of length 5 (the precise definition of prewellordering is given in $3). In this paper we study o(A) = supremum {C:

5 is realized in A}

for various A’s, in the context of a set theory that assumes full determinacy, A D (most of the time), and dependent choices for sets of reals, DC (some of the time). We are particularly interested in the projective classes A: and our most quotable theorems concern these: If AD, then for each k 3 1 , o(A:) is a cardinal. Zf AD and DC,then for each k 2 1, o(A:) 2 Kt; also for each odd k 2 1 , o(Al) is a regular cardinal and each subset of o(A:) is “ll: in the codes”, in a reasonable sense of this expression. However, our results apply to interesting larger classes of definable sets, e.g. A:, and even to the full power of R x R In §$1,2 we establish notation and give a brief summary of the properties of recursive functions, projective sets and determinacy that we need. The main new constructions of this paper are in $3, the consequences when A = R x R in $4, the consequences when A is assumed to satisfy various structure properties in $5. In §$6,7 we collect the results about the projective classes and the classes A;, respectively. I wish to thank R. M. Solovay for many discussion during the Jerusalem meeting which led to a substantial improvement of the results reported here. The problems of this paper were very much in my mind in the period immediately after my father’s death, when his loss was felt very keenly; the final manuscript was completed in his beloved Phaliron house one year after his death, to the day. For these, and for many other reasons, I wish to dedicate this paper to my father’s memory.

.

The author is a Guggenheim fellow. Work on this paper was sponsored in part by an N.S.F.grant. 24

DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM

25

$1. Preliminaries. Let o = {0,1,2, ...} be the set of natural numbers R = Ow the set of number-theoretic functions or (for our purposes) real numbers. We study subsets of all product spaces

3 = XI x

x

xk

where each X iis w or R ; we use x, y , z, w as variables over such product spaces, k, I, m , n , t , s, ... as variables over o and a, p, y , ... as variables over R . If x is an i-tuple and y a j-tuple, we let ( x , y ) be the i j-tuple whose first i components are the components of x and whose last j components are the components of y . Similarly, if X = X, x x Xi and 9J = Y , x x Y j , then

+

Xx9J = x,x

... X X i X Y , X ... X Y ] .

It will be necessary to assume a minimum knowledge of recursion theory which can be easily acquired from Kleene 1952 (or any elementary text), in particular the so-called enumeration, iteration (S,") and (second) recursion theorems. The approach to recursive and continuous partial functions on X to 9J for product spaces 3, 9J that we outline here is substantially that of Kleene 1959. As usual, ( n o ,..., nk> = 2"0+'* 3"'+'. ....p;l<+l, with pi the ith prime,

(n)i = supremum { j : p i f 1 divides n} , (a)i = Ata((i, t ) ) .

Put

Z(0) = 1 , Z(t

+ 1)

= (L%(O),...,a(t)),

ii(t) = n ,

and if x = (x,,

..., xk) with

X ~ E Wor

x~ER,

n(t) = ( n , ( t ) , . . . , a , ( t ) ) E o k .

A partial function f : X --f o is recursive if there exist recursive partial functions g(n,, ...,nk) , h(n,, ...,nk) such that f(x)

N

g(f(s)), s

N

pt[(Vt'

< t ) [h(f(t'))

N

01 & h(Z(t)) > 01.

Clearly f : 3 + o is recursive in this sense if f ( x ) depends only on a finite initial segment of the real components of x and in an effective way. A partial function f :X + R is recursive if there exists a recursive partial function g : X x o --f w such that

26

in particular

YIANNIS N. MOSCHOVAKIS

f ( 4=

Mx,t);

D o m a i n ( f ) = {x: Vt[g(x,t) is defined]}.

Finally, a partial function f :X + 'I) = Yl x ... x & is recursive if there exist recursive partial functions f l , ...,A on 2-l to w or R as required so that

fW

= (fl(X),

...,A(x>)*

We relativize this concept by introducing parameters : a partial function f : X + 9 is recursive in p (or p-recursive) if there is a recursive g : R x X + 'I) such that f(4 = g(P,x). Each product space X is a topological space, in fact a metric space, where we take o as discrete and then and each X with the product topologies. It is easy to verify that iff :X + 'I) is p-recursive and totally defined, then it is continuous. This suggests an extension of the continuity property to partial functions: a partial f : X + 9 is (partial) continuous if it is p-recursive for some p . Although the topological significance of continuous partial functions is not immediately apparent, they are a very useful class because a decent part of recursion theory can be generalized to them. Let type(X) = 0 if X is a product of copies of o ,type(X) = 1 if at least one factor of X is R . One easily verifies that if type(X) = type('I)),then there exists a recursive function n:X+'I)

which has a recursive inverse and is a topological homeomorphism. These canonical homeomorphisms give us in particular recursive pairing functions, i.e. recursive bijections on X x X to X with recursive inverses. It is not hard to prove the following characterization of continuous partial functions: If type('I)) = 0 , then a partial function f : X+ 9'J is continuous if and only if Domain(f) is open and f is continuous on its domain. If type@) = 1 , then a partial f:X + 'I) is continuous if and only if Domain(f) is a Gd set (a countable intersection of open sets) and f is continuous on its domain. The following three basic results about continuous partial functions can be deduced easily from the corresponding results about recursive functions on w to w and the definitions above. PARAMETRIZATION THEOREM(the analog of enumeration). For each X, 9 there i s a recursive partial function @: R x X + 9 such that each continuous partial f : X - 9 i s given b y

DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM

f(x)

=

27

W , X )

f o r some E E R . T o conform with standard notation of recursion theory we put @(E,X)

=

{&}(X)

{&}xqX),

N

where we shall put the superscripts in only when necessary to avoid confusion.

ITERATIONTHEOREM. For each X there is a recursive function S x :R + R such that f o r all 9,3 {El

(x, Y )

.)I

=

{SX(E,

(Y) 9

i.e. with the superscripts {E}XX993(X,y)

=

{SX(E, x)}9*3(y).

RECURSIONTHEOREM. For each X there is a recursive function F X :R+R such that if E* = FX(&),then f o r every 3 { E } (E*,

We shall be studying classes A, such class A put (1-1)

x)

2

r, A

{&*I(XI.

of subsets of R . F o r any

X, any

[X]A = { A c X: f o r some B E A and some recursive

f:X

+

R, A =f-’[B]}.

The notation [X]A is cumbersome and we shall avoid it by writing “ A EA” or even “ A is A” for A E X when it is clear (or irrelevant) which space X is involved. There is a confusion in this convention if X = R unless A is closed under recursive preimages, but all the classes we care about will satisfy this condition. The following operations on classes of sets will be useful:

&A = { A n B : A , B E A } (conjunction, &)

VA

=

{ A U B : A , B E A } (disjunction, V) 1

i A

=

{R - A : A E A } (negation or dual, 1)

XOA = { A :f o r some B E A , B s o x R , a E A o (3n)[(n, a ) E B ] } (existential number quantification, 3 m )

-

X’A = { A : f o r some B E A , B 5 R x R , a EA (3p) [(B, a ) E B ] } (existential real quantification, 3a) IIOA =

1 x 0i

II’A

i X i TA (universal real quanrwcation, V a ) .

=

A (universal number quantification, V m )

28

YIANNIS N. MOSCHOVAKIS

The classical projective classes are defined starting with

EA = { A c R : A is open} and then proceeding inductively,

n:

=

Xi+'

=

A;

172; ,

c'n: , = E:nn:.

Now A is open if and only if there is a continuous partial f : R --f o such that A = f-'[{O}]. We obtain so-called light-face classes if we start with preimages of (0) by recursive partial functions, Zk = ( A

R : f o r some recursive partial f : R + w ,

O I EA e f ( u )

zO]

and then proceed inductively with the operators C' , 1, n as before. It is well-known that the projective classes are closed under some of the operations above, e.g. 72; is closed under continuous substitution (i.e. B EEL and f : R + R continuous a f - ' [ B ] EX;), &, v , Xo, C' and for k > 0, no. We shall use these closure properties extensively, sometimes without explicit mention. A different approach to the projective classes is through definability in the language of second order number theory or analysis. The language of number theory has the usual logical symbols, number variables n, m , ..., = , the individual constant 0 and the function symbols ', +, * . For analysis we add real variables a,P, ... , so that e.g. a ( n ) , p(a(n)) are number-terms, and the corresponding quantifiers 3a,Va. By a classical lemma of Godel every recursive relation on w is definable by a formula of number theory. From this it follows trivially that every recursive relation on R is definable by a formula in the language of analysis with no bound real variables. This in turn gives simple syntactical characterizations for 72j,n:, 72:,IIi when k 0 , e.g. a set A c R is in Ei if and only if it is definable by a formula of the form

=-

3alVaZ

3a30(p0,

a,

aZ, a 3 )

where 0(P,a,al,az,a3) has no real quantifiers and Po is a fixed real parameter. One may study classes of sets which are definable in languages richer than second order number theory. Here we shall go only one step further, i.e. consider third order number theory which is obtained by adding variables F , G , H , (to second order number theory) which range over Rw, so

...

DETRMINACY AND PREWELLORDERINGS OF THE CONTINUUM

29

that F(a), P(G(a)) are number-terms, and the corresponding quantifiers 3 F , V F A subset A of R is Z: if it is definable by a formula of the form

.

3F w o , a, F )

where O(B, a, F ) has no bound type-3 variables and Po is a fixed real parameter. (If we do not allow the parameter P o , we obtain the class X:.) As usual, ll: = i Z:,A: = Z:n ll:. The classes Et, (k > 1) are defined by the obvious extension, e.g. for 72; we take formulas of the form 3 F VG O(Qo, a, F, G) . 92. Structure properties. We are interested in structure properties of classes of subsets of R or a fixed product space X. In this section we summarize a few fundamental such properties with which we shall be concerned and we outline briefly the known results about them. We let A, r, A stand for arbitrary (non-empty) classes of subsets of R and we recall the definition of [X]A in (1-1).

2.1. Universal sets. A set G E R x X is (R-)unioersal for [X]r if G E and

[xlr

=

{G,: ~ E R ) ,

where G, is the a-section of G,

G,

= (x:(~,x)EG}.

If [X]r has G as a universal set, we say that [X]r is (R-)parametrized (by GI. Suppose r is closed under recursive substitution (i.e. recursive preimages) and is parametrized by some G c R x R . For each X , put

h(x) = A t ( Z ( t ) ) ; now h : X --f R is an injection and it is easy to construct a surjection g: R +X which agrees with h-' on the image h[X]. In particular, for each B E X, h-'g-'[B] = B . From this it follows that each B E [X]r is of the form h-'[A] for some A E (where ~ A = g-'f-'[C], if B = f -'[C], with recursive f), and hence

(2-1)

H = ((a,x ) : (a, hx) E G }

parametrizes [X]r. One then proceeds to show easily that for such r, i r , COT, C ' r , nor,II'r are also parametrized. It is a classical result that the projective classes Z,,!, ll: are parametrized.

YIANNIS N. MOSCHOVAKIS

30

We can prove it here by taking for CA the set G = {(&,a): { & } ( a ) = o}

and then proceeding inductively by the procedures above. A simple diagonal argument shows that if r is closed under recursive substitution and G parametrizes I-, then G 6 i r . Thus for each k , EL- II; # Id, and we obtain easily the hierarchy theorem, 2.2 Separation and reduction. r satisfies the separation property if for each A , B E r with A n B = Id, there exists C E r n i r with A E C, B n C = Id. r satisfies the reduction property if for each A , B Er there existAl,B,ErwithAl E A , B , E B , A , n B , = I d a n d A , U B , = A U B . One easily verifies that Reduction(T) 3 Separation (ir)- to separate two sets reduce their complements. A slightly trickier argument shows that if is parametrized and closed under recursive substitution, then Reduction(T) 3 not Separation (r).Thus under these hypotheses reduction cannot hold for both and i r and if it holds for one of these classes then separation holds for the other. It has been known from the classical work in descriptive set theory that CA, II: and Cl satisfy the reduction property. Whether reduction holds on the C or the II side for k >= 3 is one of the central problems of the theory of projective sets and has provided much of the motivation for their study. The best result until very recently was in Addison 1959a, 1959b (working out a proposal of Godel, 1940) that if every real is constructible in the sense of Godel, then Reduction (Ci) for all k 2 3 . 2.3 Determinacy. With each subset A E R x R we associate a game as follows: players I and I1 choose successively natural numbers a(O), p(O), a ( l ) , p(1), ... and if (a, p) E A , I wins, if (a, p) $ A , I1 wins. A strategy for player I is a real a (utilized as a function on finite sequences of integers to integers) which tells I how to play when 11 plays any real p. We let a * [,4] = a where a(n) = a(P(n)).

Similarly for player 11 we put [a]

*T

=

p where p(n) = r@(n + 1)).

A strategy r~ is winning for I if for all p, (a* [PI, p) E A ; z is winning for 11 if for all a , (a, [ a ] * z) $ A . The set A is determined if either I or I1 has a winning strategy-it is trivial that they cannot both have winning strategies. Infinite games of this type were introduced in Gale-Stewart 1953, where

DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM

31

it was shown that every closed set is determined and that there exist nondetermined sets. The relevance of determinacy for definability problems in analysis was emphasized by Mycielski and Steinhaus in the early sixties, cf. Mycielski 1964 for references. The proof that non-determined sets exist utilizes the axiom of choice, probably in an essential way. Hence Zermelo-Fraenkel set theory (without choice) may be consistent with the following proposition. Axiom of Determinacy, A D . Every subset of’ R x R is determined.

Fortunately AD implies (trivially) the Countable axiom of choice f o r sets of reals, Vn3a(n,cr) E A + 3aVn(n, (a),)

EA

( A E o x R)

which is indispensable in analysis. AD also implies several “desirable falsehoods”, e.g. that every set of real numbers is Lebesgue measurable and (closer to our subject) that every wellorderable subset of R is countable. Such results led Steinhaus to propose that we replace the axiom of choice by A D in our thinking about sets. A more persuasive incentive for studying consequences of A D is the hope that we may find a natural class of sets which contains R and satisfies A D this was first suggested by Mycielski. Recently Solovay has been conjecturing that L [ R ] = the class of sets constructible f r o m R

satisfies A D and that this may be provable from suitable axioms of infinity. A very powerful argument for Solovay’s conjecture is the recent theorem of Martin: if there exist measurable cardinals, then every II: set is determined. (However a theorem of Silver together with the results mentioned in 2.4 below shows that the existence of measurable cardinals does not imply that every A; set is determined, unless through an inconsistency.) Addison-Moschovakis 1968 proposed that every subset of R x R ordinal dejinable f r o m a real is determined. At this time we have no lead for attempting to prove this assertion from plausible axioms, so we must view the proposal only as a challenge to derive consequences from it that contradict our basic intuitions about sets. Sometimes we can obtain interesting results by assuming that only sets in a certain class A are determined. Put Determinacy (A): every A E A is determined,

where as usual A E A for A E R x R means A E [ R x R I A . Consequences of Determinacy(A) are particularly pleasing when the sets in A are definable so that the hypothesis is plausible.

YIANNIS N. MOSCHOVAKTS

32

2.4 Prewellorderings. A prewellordering of a set A (with field A ) is a relation 5 on A such that for some ordinal 5 , some surjection 4 : A + 5 and all x, y E A , x

5Y *

5 4(Y),

where on the right 5 is the ordering relation on ordinals. It is immediate that such 4 and 5 are unique when they exist; we call 5 the length of 5 and 4 the canonical surjection (of 5 ) . A relation 5 is a prewellordering if and only if it is reflexive, transitive, connected and well founded - from being a wellordering it only lacks antisymmetry. It often happens that we can prove the first three of these properties for some 5 , but instead of well foundedness we only have the apparently weaker lack of infinite descending chains, i.e.

Vn[xn+, 5 xnl

3n[xn 6 xn+1] s

=>

where { x ~ } . ” =varies ~ over all sequences in A . In order to infer that 5 is a prewellordering, we then need the axiom of dependent choices - in our case, when A E X for some X, the following version for sets of reals is enough:

Dependent Choices, DC. For each A E R x R , V U ~ P P) ( ~E,A

=>

3aVn((U)n, (a), + 1) E A

.

This follows from the axiom of choice and it is not known whether it follows from A D , or whether it is consistent with A D . However any class of sets that contains R must satisfy DC, since DC holds in the universe, hence any “natural” model of A D (hopefully L [ R ] ) will also satisfy D C . Consider the following property of a class r parametrized by G .

Prewellordering (I-). There is a prewellordering 5 on G and relations - and 5 in r and i r respectively, such that 5

(2-2)

W E G => V z { z 5 w

If G’ is also universal for

z

2w

e [ z ~ G & z5 w ] } .

r , then

for some recursive f and some y o . If we define (El,

PI> 5 ‘ (UZY P 2 ) * ( r o J (UlY P I ) ) 5 ( Y o J (U2Y P 2 ) )

Y

we can easily verify that the conditions of Prewellordering (r)are satisfied with 5 ’ and suitable 5 ’, 3 ’ . Thus the prewellordering property depends on alone and not on any particular universal set.

DETERMINACY A N D PREWELLORDERLNGS OF THE CONTINUUM

It is easy to verify that if substitution and &, then

r is parametrized

Prewellordering

33

and closed under recursive

(r) e- Reduction (r).

Actually Prewellordering (I?) implies many interesting structure results about r , especially when r satisfies sufficiently strong closure conditions. One may say that (1)

Prewellordering (ZJ),Prewellordering (II:)

were known classically. (Actually Prewellordering (ZA)was only noticEd by Addison in 1968.) We formulated the prewellordering property in 1964 in order to prove Prewellordering (Xi)and thus lift in an elegant manner the theory of II: to Z;.(One of our results was the construction of a hierarchy for [o]Ai, which we then found had been achieved by Suzuki 1964.) This construction appeared in Rogers 1967 where it was shown (in effect) that

(11) i f r i s parametrized, closed under recursive substitutions, &, v, Vcc and Prewellordering (r),then Prewellordering (C'T) The prewellordering property was the key to the development of the theory of semi-hyperanalytic sets in Moschovakis 1967 and the theory of semi-hyperprojective sets in Moschovakis 1969. In each of these cases we can lift much of the theory of II: to these classes - and we can do more because of the stronger closure properties that we can utilize. The arguments of Addison, 1959a, suffice to show

.

(111) if r i s parametrized, closed under recursive substitution, &, v, 3m,Vm and Vcc and some relation in r wellorders R with order type K,, then Prewellordering (C'T).

These results imply that if every real is constructible in the sense of Godel, then for each k 2 2 , Prewellordering (Xi). Determinacy enters this picture via the next theorem, the main result in Addison-Moschovakis 1968 and Martin 1968. (IV) If

r

i s parametrized, closed under continuous substitution, &,

Prewellordering (r),D C a n d Determinacy (rni r ) , then Prewellordering (II'r).

V,3m, Vrn and 3a, if

If we assume DC and the determinacy of all projective sets, then (I), (11) and (IV) imply Prewellordering (Xi)for all even k and Prewelfordering (II;) for all odd k . Since the prewellordering property cannot hold on both the X and the II side for the same k (because it implies reduction), this picture of the projective hierarchy is radically different from the picture when we assume that every real is constructible. Which is the correct picture is

34

YIANNIS N. MOSCHOVAKIS

perhaps not absolutely clear yet, but it is fair to say that many people working in this area and prone to speak about truth in set theory (ourselves included) tend to favor the alternating picture. In fact the most persuasive argument for accepting projective determinacy (aside from Martin’s proof of Determinacy (II:)) is the naturalness of the known proofs of (IV), both Martin’s and ours. One of the central open problems in the theory of definability on the continuum is whether the prewellordering property holds for X : or IIf. Trivial extensions of the method used for (111) show that if every set of reals is in L [ R ] , then Prewellordering ( X I ) , but we are interested in answering this question using axioms that do not restrict our conception of set. It is not unreasonable to suppose that Prewellordering ( X i ) may be provable in Zermelo-Fraenkel.

53. The basic lemmas. Let A be a class of subsets of R , to avoid trivialities assume that A contains all singletons and is closed under continuous substitution, & and i . An ordinal [ is realized in A if there is a prewellordering S of some subset of R in A (i.e. in [ R x RIA) with length [. The assumed closure properties of A imply

5 is realized in A & < t * 5 is realized in A , [ is realized in A a [ [>0

&c

+ 1 is

realized in A ,

is realized in A 5 5 is the length of some prewellordering of R in A .

It is also clear that the same ordinals are realized if we allow prewellorderings on subsets of any product space X . Put 0th) = supremum {[: 5 is realized in A}.

We wish to study o(A), especially when A satisfies nontrivial closure conditions, e.g. A = Ai,A:,R2. The basic theory in which we work is Zermelo-Fraenkel without choice but with the countable axiom of choice for sets of reals. All results stated thus f a r in this paper a r e provable in this theory. We shall often assume A D and sometimes D C , but then will mark the theorems accordingly. To verify that o(A) is always defined even without the axiom of choice, notice that otR2) = supremum {[: there exists a surjection ~ : R - B [ ) =

supremum {[: there exists a n injection

+:t + ‘2);

now the second class in braces is bounded by a classical argument of Hartog. In this section we give the basic new constructions of this paper.

DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM

35

A formula of the language of analysis is X; if it is of the form (3-1)

3a,3a2

... 3a,8,

where 8 has no real quantifiers. These are the formulas from which we get the Et subsets of the product spaces X , by fixing the values of some of the real variables and considering the remaining free variables as a “vector” variable over X . Up to equivalence in the standard interpretations these formulas satisfy all the closure properties of Ct sets. Let j ( be a partial function on R to o . We extend the language of analysis by adding prime formulas of the form “ ~ ( 6 )N m” , which in the standard interpretation mean “ ~ ( 6 )is defined and = m” . Now Zi(x) formulas are those of the form (3-1) where 0 has no real quantifiers and only positive occurrences of subformulas of the f o r m ~ ( 6 N ) m . We let &(x) be the collection of subsets of R which are definable by X : ( x ) formulas of this type, after we fix the values of all the variables but one real variable. If x is totally defined, then the restriction to positive occurrences of subformulas of the form ~ ( 6 N ) m is not essential, since i ( ~ ( 6 )N

m ) o 3n[n # m &x(S)

N

n].

LEMMA1. For each Zt(x) formula 0 there i s a Z: formula Y(a,P) (with no occurrences ofx) such that

0 * 3a3P{Vn[x(a),

(3-2)

=

P(n)l 8z ‘y(a,P)}

9

i.e. the universal closure of (3-2) i s true in the standard interpretation.

Proof. It is enough to prove the lemma for 0 that have no real quantifiers, since the result then follows by simple quantification. These are built up from formulas with no occurrences of x and formulas of the form ~ ( 6 2): m by the positive operations &, v , 3 m , V m , and the proof is by induction over this construction. One of the basis cases is handled by

~ ( 6N ) m

3 a 3 P { V n [ ~ ( ( a ) , )N P(n)] 8c

= 6 8c P(0) = m }

.

To treat the most complicated case in the inductive step, suppose 0 o Vt@*(t) , where by induction hypothesis, @*(t>

* 3 w q V n [ x ( ( a ) , ) = P(n)l 85 Y * ( %P, t>>

*

Now the countable axiom of choice for sets of reals implies that Vt@*(t)

* ~ r ~ ~ { v n [ x ( ( r=) ,6(n)l > V t 3 ~ ~ P { V n C ( a= ) n (Y)I &k W P ( n ) = s ( ( t , n > > l

Y * ( %P,

o>> Y

YIANNIS N. MOSCHOVAKIS

36

from which the desired conclusion follows by using the closure properties (up to equivalence) of formulas. From this lemma we obtain immediately LEMMA2. Let G c R4 be universal f o r [R3]3c:, the ternary relations in x;,f o r each partial x: R + o put

(3-3)

(4@ E G(X)

* 3a3P{Vn[x((a),) 21 P(n>I & ( E , 8, a, P ) E G ) ;

then G ( x ) is universal f o r Z;(x).

Let S be a relation with field a subset of some X with type 3E = 1 , let n : R + X be the canonical recursive homeomorphism. The canonical partial 1:R -, w associated with 5 is the characteristic function of S , restricted and carried over to R for convenience, to the domain of (3-4)

S ~ ( L x&) n~ = 01

~(aN ) n e-

v [n(a)o S n(.>o

& nwl

&

S

n(co1

S 71(a>,l

1 >.o(.[

& n = 11.

We shall be concerned with the class

(3.5)

W S )

=

x;w,

for this canonically defined x, especially when is a prewellordering. If the length of 5 is 5 and 4: F i e l d ( S ) -+ 5 is the canonical surjection, then 4 gives a “coding” of 5 in the space 3 , i.e. we can think of each x ~ F i e l d ( S such ) that 4(x) = q < 5 as a code or name for q. 1ff:c + 92 is a function on 5 to subsets of some 9, we can represent it by a subset of X x ‘1) as follows: C o d ( f ; 4 ) = { ( x , ~ ) :x

5 x&yEf(4(X))}.

Suppose f:5+32 is a function. A choice subfunction o f f is any g : ( + 92 such that for all q < 5 , g(?) E

f(?)7

* g(v) f 2. The interesting case is when for each q < 5 , f ( q ) # f(?> #

a nonempty subset of each f ( q ) .

65, when g(q) “chooses”

LEMMA 3 (Main Lemma). Assume A D . Let S be a prewellordering with field a subset of some X and length 5 , let f : ( + b 2 be a function.

DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM

37

Then there exists a choice subjunction g o f f such that Cod(g; 5 ) is a E l ( 5 ) subset of 3E x '2). Proof. For each [ 5 5 , let f c be the restriction o f f to [, modified to give Pr on 5 - [ , fc(1)

=

f(1)

= @

if 1 < i if [ j q < t .

Suppose there is some [ S 5 such that f c does not have a choice subfunction with Cod in Ei ( 5 ), let I be the smallest such 5 . The lemma will be proved if we can deduce a contradiction from this assumption. Let 4 : Field ( 5 )+ 5 be the canonical surjection. First we argue that I is a limit ordinal; because if 1 = [ + 1 and gc is a choice subfunction of f c , then either f([) = Pr and gc is a choice subfunction of f, or there is some y o Ef ([) and

g,

=

is a choice subfunction of (x, Y ) E Cod(g,;

gc - { ( C Y @ ) } fA

"

{(CY

{YODI

with

5 ) * (x, Y ) E C o d k c ; 6 ) v C4W

=

5

Y = Yo].

If xo E Field ( 5 )is chosen so that 4(x0) = [ , then $(x) = [

-=. x 6 x,&x,

j x,

so that Cod(g,; 6) is in Z; ( 5 )contradicting the choice of 1. By Lemma 2, Zi ( 5 )is parametrized, hence the class [X x '2))IE:( 6 )of Z: ( 5 )subsets of X x '2) is parametrized, let G E R x 3E x '2) be a fixed universal set for it. As usually, G , = { ( x , Y ) :( ~ , ~ , Y ) E G } . Consider the following two person game. If I plays CY and I1 plays B ythen

I1 wins e

i (3q)[g,

v ( 3 1 < A)@[

is a choice subfunction o f f , & G, = Cod(g,; 51

c 1)(3g,)(3gC)[g,is a choice subfunction off,

& g c is a choice subjunction of f c & 1 < & G, = Cod(g,; 5)& G, = Cod(gc; S)].

If we think of u as a code of a function g when G, = C o d ( g ; g ) , then 11 wins if either I does not code a choice subfunction of an initial segment off or I does, and I1 codes a choice subfunction of a longer initial segment of j .

YIANNIS N. MOSCHOVAKIS

38

By A D , the game is determined.

Case 1. I has a winning strategy

0. Now

for each

p there is some

q = q(p) and some gvts,, a choice subfunction of fvt,,, so that

GO*[,, = COd(gq(,);9

-

If supremum {q(/?): p E R } < [ < A for some i, then I1 can win against this 0 by playing p so that G, = Cod(g,; I), for some choice subfunction g, of f, - such a p exists by the choice of 2 . Since 2 is limit, the other alternative is supremum {q(fi): /? E R } = A . Put now g, is clearly a choice subfunction of f , and

( x , ~ ) ~ C o d (5 g ,)~* ; ~P[(~*[P],x,Y)EG], which implies that Cod(g,;x) is in Zt(6) contradicting the choice of A .

Case 2. 11 has a winning strategy the set G 3 x $9defined by (x, y ) E

o

T.

For each E E R ,W E X , consider

w I w & 3z[z S w & i ( w 5 z ) & { E } ( z ) i s defined 82

({El

(z), x, Y ) E GI 7

where { E } ( z ) N { E } ~ , ~ ( Z i.e. ) , we think of E as a code for a continuous partial function on X to R . It is clear from the closure properties of E i ( S ) that each Ae,wis in X:(S)-notice that i ( w 4 z) c> 3ct[n(~),, = w & ~ ( c 1 ) ~= z & x(a) N 13 when x is the canonical partial function on R to o associated with , and w w, z 5 w , i.e. z E F i e l d ( 5 ) . In fact, there is a recursive function g : R x X R such that for each E , W , A&,W = G g k w ) = { ( x ,Y ) : (g(E, w), x, Y ) E

GI *

This follows by recalling that G was defined explicitly by a Xi(x) formula and that the closure of Zi(x) under the operations &, v, ~ C etc. X, follows from simple explicit manipulations of X:(x) formulas which can be easily made to correspond to recursive operations on the codes. The recursion theorem for continuous partial functions implies that there is a fixed real E* such that for all w EX,

{&*I (w)

21

ME*, 4 1*7

5

where the strong equality _N is the same as = here, since g is totally defined on R x 3 , so that [g(E*, w ) ] * T is always a real.

SUBLEMMA. For each w E F i e l d ( I ) there i s some choice subfunction g,(,) of frcw,such that

5

= I;(w) and some

DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM

39

and Proof of the sublemma is by transfinite induction on 4 ( w ) . If both of the assertions hold for all z E Field ( S ) with 4(z) < 4 ( w ) , then easily

A&*,,

u

=

C o d k , , ; 51,

=

where and for [ < q ,

{Cod(g,(,);3 :4 b ) < 4(w)l

q = supremum { [ ( z ) : 4(z)

g,K)

< +(w)}

u {&(,)(o:

4(z> < 4 < w > > .

=

Clearly g,, is a choice subfunction of f,,;hence if q = 2 , we have already obtained a contradiction, since is Zi(S). If q < 2 , we have in any case that q 2 4 ( w ) , since by induction hypothesis for each z with 4 ( z ) 4 ( w ) 5(z) > 4 ( z ) * Now

-=

Y

A&*,, = GE(&*,W)- C o d k , , ; 5 ) and the choice of z implies that ‘[g(E*,W)]*T

‘Od(g[; 5 )

=

for some 5 > q 2 $(w) and some choice subfunction g c of fr. Since [g(e*,w)] * z = { E * } ( w ) by the choice of E * , the proof of the sublemma is complete. If we now put

gd5)

=

u

WE

Field(

S)&(w,(O

9

then easily gA is a choice subfunction of fA and

( x , y ) ~ C o d ( g L5; ) *

6 w & ( { E * } ( w ) , x , Y ) EG I ,

~ W [ W

so that Cod(gA;5 ) is in Zt(5) contradicting the choice of ;i and completing the proof of the lemma. Let 5 be a prewellordering with length 5 and canonical surjection 4 : F i e l d ( 5 ) + 5. Let f : 5’’ -,‘4’2 be an n-ary function on 5 to subsets of ‘1). By analogy with unary functions put Cod(f;5 ) =

{(XI, * * xn,

Y):~

1*,

xn E

Field( 5 ) & Y ~ f ( 4 ( ~ 1 * * ) ,4 ( x 3 ) } *

5 and any subset A of 5 , put .. ., x, E Field( 5 ) & P ( 4 ( x l ) ,...,$(x,))},

Similarly for any n-ary relation P on

Cod(P; 5 ) = {(x,,

.. .,x,):

Cod(A; 5 )

=

xl,

{ x : x E F i e l d ( g )& ~ ( x ) E A } .

YIANNIS N. MOSCHOVAKIS

40

I f f : 5" -92, then a choice subfunction off is any g : 5'' -+ s2 such that for all ql, g,, < t , g(q1, * * * q n ) 5 f ( ~ l , * * . , ~ nf )

f

-. V n ) ;

( ~ 1*,

9

d V l ? . * * , V n ) f @.

-+

LEMMA4. Assume A D . Let 5 be a prewellordering with jield a subset of some X and length 5 , let f : 5" -+ g2 be a function. T h e n there exists a choice subfunction g o f f such that Cod(g; 6 ) is a Zi(S) subset O f X X O .

Proof. We know the lemma for unary functions, so to proceed inductively let us assume it for n-ary functions and let f be an n 1-ary function on 5 to 92. For each g < 5 , put

+

f

and set f*(g)

=

" ( ~ 1 ,*

*

qn)

=

f

( ~ 1 * ,*

qm

{ a : f o r some choice subfunction g" off", G, = C o d ( g " , S ) ) ,

where G is universal for the Z;(S) subsets of (X>n x '1). By Lemma 3, f * has a choice subfunction g* with Cod(g*; 5) in E i ( 5 ) . Moreover, if (xl,.

. . , x n , x, y )

* 3a[(x, 4 E Cod(g*; S ) 8~(a, X I , . . .,x,,,Y>E G I ,

then A is Ei(S) and it is easy to verify that A

=

C o d ( g ; S)

for some choice subfunction g o f f . LEMMA5 . Assume A D . Let S be a prewellordering with jield a subset of some X and length 5 , let P be an n-ary relation on 5 , A a subset of 5 . Then Cod(P; S ) , Cod(A; 6 ) are Z i ( S ) . Proof. The part about relations follows easily from Lemma 4 if we associate with n-ary P on 5 the function f(ql,..*,gn>

= {ao)

if f'(ql,**.,vn),

= ( ~ 1 )

if l P ( g 1 , * * * , q n ) ,

where a,,,al are two fixed distinct reals. Now the only choice subfunction o f f is f itself, so by Lemma 4 C o d ( f ; S ) is Zi(S) and (xl,

...,x , ) ~ C o d ( P ; s )0 ( x l ,...,x , , a , ) ~ C o d ( f ; 6 ) .

The part about sets follows trivially from that about relations. One interesting application of Lemma 3 is given in the next lemma.

DETERMINACY AND PREWELLORDE RINGS O F THE CONTINUUM

41

LEMMA 6 . Assume A D . Let 5 be a prewellordering withjield a subset of some X and length t , let x be the partial function associated with - via (3-4). Let r be a class of sets containing all singletons, parametrized I and closed under continuous substitution (preimages), &, v, 3m, tlm and 3a and containing ( ( 6 , m ) : ~ ( 6 E ) m } . Suppose A = where each A,, is a

Proof. Let G of ‘2). Put

Uq<&

r subset of some ‘2).

E

Then A E ~ .

x ‘2) be universal for the class

f(V)

= { a : G, =

[‘2)]rof r

subsets

A,}

and by Lemma 3 choose g: 5 -+ R 2 , a choice subfunction o f f such that C o d ( g ; S ) EX:( S ) . Now the closure properties of r imply that C o d ( g ; S)Eland we have y E A o 3x3a[(x, a) E C o d ( g ; 5 ) & (a, y ) E G I ,

so that A E r . 54. How large is 0 ( ~ 2 ) In . set theory with the axiom of choice 2”’ is an aleph and

0 ( ~ 2 )= (2”’)’

= jirst aleph greater than 2 ” O .

A D implies that R cannot be wellordered so that 2”’ is not an aleph and the formula above makes no sense. However 0 ( ~ 2surely ) is an aleph and it seems to give a reasonable measure of the size of R relative to the size of ordinals. It turns out that in the context of A D , 0 ( ~ 2is) utterly huge. In this section we give a few results that suggest this. The first answers a question of Solovay.7 Results of this type were first obtained (to the best of our knowledge) by H. Friedman and ourselves, independently, in the winter and spring of 1968. The main result at that time was that (with A D ) many alephs which can be approached from below (e.g. nxl) are order types of prewellorderings of R. Friedman’s results gave additional information about the subsets of these alephs, e.g. that each subset of nxI is definable from a real. Our chief result was that (with A D and DC) each uk is the order type of a projective prewellordering of R and that larger ordinals, like n,, are order types of hyperanalytic prewellorderings. In our original proofs we only used Solovay games, like that in the proof of Theorem 6, i.e. games whose definition guarantees that player I cannot win; such games were first in the codes”. utilized by Solovay in his proof that (with AD) every subset of uI is Friedman’s early contribution was a technique of utilizing games where it is not clear which player wins, by proving the desired result by cases; we call games used in such proofs Friedman games. The proof of our key Lemma 3 uses Friedman games and was obtained after we learned this technique from Friedman; we like to think of it as the product of the marriage of Friedman’s technique with our method of introducing “coded classes of sets” and applying the recursion theorem. Friedman’s results will appear in a joint Friedman-Solovay paper with tentative title “Large ordinals and the axiom of determinateness”.

“n;

42

YIANNIS N. MOSCHOVAKIS

THEOREM 1. Assume A D . I f there is a surjection 4: R is a surjection 4*: R + { 2 . (The hypothesis is trivially equivalent to 5 < 0 ( ~ 2 )for

Proof. Assume defined on R by

5, 5

then there

# 0.)

4: R .+ 5 , let 5 be the prewellordering with length 5

+

5B *

5 +(PI. If A c 5 then by Lemma 5 C o d ( A ; 5 ) is E ; ( s ) ; thus if G is a universal a

+(a>

set for Z i ( s ) we can define a surjection +*(a) =

=

+*

by

if G, = C o d ( A ; 5 )for some A

A

E

t,

Pr otherwise.

COROLLARY 1.1. (Friedman) Assume A D . I f 5 < 0 ( ~ 2 )then

5' < 0 ( ~ 2 ) .

Proof. Using Lemma 5 again, but the part about binary relations, we ) there is a surjection t+h : R -+ct;xe)2. show as in the theorem that if 5 < 0 ( ~ 2then + 5' given by But there is an obvious surjection x(A) = order type of A , if A =

E

5

x 5 is a wellordering,

0 otherwise,

so there is a surjection x

o

+ 5' , hence 5' < 0 ( ~ 2 ) . A D . I f there is a surjection 4: R+,5

$: R

COROLLARY 1.2. Assume

which

i s ordinal definable j r o m a real (in L [ R ] ) then there is a surjection

$: R + { 2 which is also ordinal dejinable f r o m a real (in L[R])sothat each subset of 5 is ordinal definable from a reaI (in L [ R ] ) . (The part about L [ R ] is due to Solovay.) Proof is immediate from the proof of the theorem since the t+h we defined is ordinal definable from a real (in L [ R ] ) when is. The best lower bounds for 0 ( ~ 2 )that we knew before the Jerusalem meeting were cardinals I which are nA in the sense of Mahlo. At the meeting Solovay told us of his results that the first Mahlo, the first fixed points of Mahlos, etc. are less than 0 ( ~ 2 ) We . prove here a theorem which gives these results (and apparently more) and which will also yield definability estimates in 67.t t Our original proofs that AD implies the existence of large cardinals (inaccessibles of high order but not Mahlo) used recursion theoretic techniques and in particular the theory of hyperanalytic and hyperprojective subsets of R, in Moschovakis 1967, 1969. The proof of Theorem 2 below uses a method that we learned from Solovay in Jerusalem: to show that a cardinal with a certain property exists and is less than o(R2), assume that no 6 < x has that property and show that we can then code all ordinals less than x using reals. Again, we like to think of our Theorem 2 as a child of coupling this technique with our coding methods embodied in Lemma 3. Solovay's applications of this technique will appear in the Friedman-Solovay paper mentioned in the preceding footnote.

DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM

43

Let L be a denumerable first order language, perhaps with many sorts of variables, among which we distinguish one, the ordinal variables. Suppose that to each ordinal A we have assigned an L-structure in which the ordinal variables range over the ordinals less than A . If A E A, we let ('&,A) be the structure obtained from 'uAby adding one set (unary predicate) of ordinals - let L' be the language suitable for these structures. If A < K , A E A and B E K , put

aA,

(1,A )

<5(K,B )

if for every formula B(x,, ...,x,) of L' whose only free variables are ordinal variables and every n-tuple cl, ..., 5, or ordinals less than A ,

(%,,A)

eel, ..., 5,)

1

wl, ..., 5,).

(From this follows that A = An B , but it need not be the case that is a substructure of !!IK.) We say that K reflects with respect to L and {'?IA} if for each A E { A : (A, A n A) << ( K , A ) } is non-empty and unbounded in K .

'uA K ,

{aA}

THEOREM 2. Assume AD. Let a denumerable L a n d a class of structures be given. There exists a regular cardinal K < 0 ( ~ 2 which ) reflects with respect to L and

{aA}.

We first prove a lemma. LEMMA7. Let x1,x2 be partial functions on R to w , assume x1 E x 2 , i.e. x1 is a subfunction of x 2 . If G(x) is the canonical universal set for X i ( x ) subsets of X dejined by (3-3) and (2-1). then G(xl) E G ( x 2 ) .

Proof. We defined G ( x ) first for X i ( x ) subsets of R by a fixed Z:(x) formula in (3-3) and then we passed to other X's by (2-l), via recursive preimages. The lemma follows trivially from our requirement that X : ( x ) formulas have only positive occurrences of subformulas of the form ~ ( 6 N) m , so that when they are true for some x, they are true for all extensions. Proof of the theorem. Let us extend to arbitrary finite sequences of ordinals the canonical wellordering on pairs of ordinals,

YIANNIS N. MOSCHOVAKIS

44

i.e. we order sequences first by the maximum, within the same maximum by the length and within the same maximum and length lexicographically. Now the predecessors of each sequence form a set, so there exists a unique order preserving map F on the class of all ordinals to the class of all finite sequences of ordinals. It is easy to verify that when 5 is regular, 5 > w , then F restricted to 5 gives a bijection of 5 with all finite sequences of ordinals < 5 ; we alter F slightly so that it has this property also at w . Let L, be given, assume K is an ordinal such that no regular 5 < K reflects with respect to L, { % A } ; the theorem will follow if we can prove

{aA}

K


, for We define by transfinite induction a mapping 4 : ~ + ~ 2where each 5 < K we think of the elements of 4(5) as the codes for 5 . If 4 has been defined for all q < 5 , where 5 5 K , put

* (3rl<5)(31:<5)[ccE4(?)&PE4(r)&? It will turn out that S Kis a prewellordering of length K . For each regular 5 < K , each A E 5 , put g5tP

T h ( 5 , A ) = { q < (: F($ = ( m , t l , ...,(,) of some formula

551.

and m is the Godel number

'4xl, ..., x,) of L' and (91t,A) b W1,..., t,)}.

It is immediate from the remarks about F above, that if q < 5 , q also regular and B c 7, (q,B)

<-= (

5

7

4

*

Th(?,B) E W 5 , A ) .

Let n:w x R x R + R be the canonical recursive homeomorphism. Each ( < K must fall under one of the cases below.

5

=

0. Put

Case 2 . 5

=

5 +1.

Case 1.

Case 3.

4 ( ( ) = (n(l,a,a): ~ E R } . Put

4(5) = {71(2,a,cc): cc~4(4')).

4 is a singular limit ordinal. Put

c

[ < 5 and some cojinal m a p f : + 5 , cc E

+(() = {n(3,~,/3): f o r some = C o d ( g * ; &),

$(c) and G,(St)

where g* is a choice subfunction of the function f *(q) = 4 ( f ( q ) ) } .

Here G(St;) is the canonical universal set for the class of E:(Sr) subsets of R x R , where Z i ( 5 ) is associated with each relation 5 via (3-4) and (3-5). As usually G,( 5 [) = ((7,s): (p, y , 6 ) E G( S ,)} . Case 4. 5 is a regular limit ordinal. Put +(C) = {n(4,a,P):f o r some c 5 and some 5 < 5 there is no A 2 5 so that (A, A nA) << ( 5 , A ) and a E 4(5) and q?(5<) = C o d ( W 5 ,4;St)}. A

DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM

45

Here H ( &) is the canonical universal set for El( St) subsets of R . We now prove by transfinite induction on 5 < K that: )E 4(5) then 5 falls under Case n , If n(n, LY, j (i) (ii) 4(5># @, (iii) r] < 5 G- 4(r])n4(5) = @. The induction hypothesis implies immediately that 5 is a prewellordering for each ( < 4 , so we can apply the lemmas of $3. Now (i) is trivial, (ii) is trivial when 5 falls under Cases 1, 2 and follows immediately from Lemmas 3, 5 when 5 falls under Cases 3, 4 and (iii) is trivial when 5 falls under Cases 1, 2. Proof of (iii) when 5 f a l l s under Case 3. Suppose n(3,cc,P)~4(1]) n 4(5), for some r]< 5, we must derive a contradiction. By (i) of the ind. hyp., r] also falls under Case 3, and using (iii) of the ind. hyp. on LY we conclude the following: there is some [< r] such that LY E 4(()and cofinal maps f,: [-+r], f 2 :( - 5 such that GP(5 = Cod(g:; 5 i) = Cod(gf; 5 c), where g:, g: respectively are choice subfunctions of the functions f :(v) = 4 ( f l ( v ) ) ,f T ( v ) = 4 ( f 2 ( v ) ) .Hence g r = g ; , since these functions are completely determined by Cod(g:; sc) C o d ( g 2 ;Sc), hence for each v < (, f : ( v ) nf2(v)# @, i.e. 4(fl(v)) n 4 ( f 2 ( v ) )# @ so by (iii) of the ind. hyp. f l ( v ) = f 2 ( v ) . This implies r] = 5 , contradicting r] < 5 . Proof of (iii) when 5 f a l l s under Case 4. Suppose 4 4 , a, p) E 4(r])n 4(5) with r] < 5 , we must derive a contradiction. By (i) of the ind. hyp., r] is regular, there is some ( < r] with LY E 4(() and there are sets A E r ] , B E 5 so that Hp(S r ] ) = C o d ( T h ( A , r ] ) ;ZV), H/?(St) = C o d ( T h ( B , S ) ;St).

Let

x,,,x,: be

the partial functions associated with the prewellorderings via (3-4). Since S V is an initial segemnt of &, we have x,,E xt so that by Lemma 7,

&, St

Cod(Th(r],A);5 J E C o d ( T h ( 5 , B ) ;St).

But since

S V is

an initial segment of Th(rl,A)

&,

this implies immediately that

c Th(5,B),

so that ( r ] , A ) << (5, B ) contradicting the definition of 4(5). The dyadic second order language f o r one ordering has individual variables, relation symbols 5 , = and E , variables over subsets of the domain of individuals (unary relations) and variables over sets of pairs of individuals (binary relations). For each ordinal A , 2, = (A;'2, ("""2, 5 , ~is) the standard model of this language.

YIANNIS N. MOSCHOVAKIS

46

We define the Mahlo numbers in the usual way.

m,

=

mA+l=

{t:5 is r e g u l a r } , {t: each closed, unbounded A c 5 intersects m,},

V1lA

=

n,,
COROLARY 2.1. Let L be the dyadic second order language f o r one ordering, the class of its standard models a s aboce. If K is regular and reflects f o r this L, then K E ~ J I , . Thus if we assume AD, there i s some K < 0 ( ~ 2 such ) that K E ~ X , .

{aA}

{aA}

Proof. Wellorder the class {(q, 5): q, 5 ordinals} first by the maximum and then lexicographically, let G(5) be the 5th pair, let ( q , 5 ) be the mapping inverse to G , so that

G((? 5 ) )

=

(V,O.

Clearly G and its inverse are definable in Land G restricted to any regular ordinal 5 gives a bijection of 5 with 5 x 5 . If A , B are subsets of K , put A * B = {(q,O): qEA}

u {(q,l):

VEB}.

If K is regular and (A, ( A * B ) n A ) < % ( K , A * B ) , then it is immediate that (A, A nA ) ~ < ( K , A ) , (A,B nA)<<(K, B). Similarly, if 5 < K and {A,,},,
*

{An},<, = { ( v , C ) : V < ~ & ~ E A ~ } *

Again it is easy to verify that if K is regular, A > 5 and (A,*{As},,
(A,( A * DI,)nA) <4 ( K , A * %I,). By the remarks above, (A, A nA) << ( K , A ) , hence A E A . Also (A,YJl, nA)

DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM

47

Since K ~m,+ the sentence “every closed unbounded set intersects !IJl,” holds in (91K,Zm,), hence it holds in (211,!IJl,nA) and A E ! I J ~ , + ~ . Thus every closed unbounded A E K intersects !Ill,+, , hence

<<(K,!IJ~,). KE!Bic+2.

(4)

If

t

is limit,

C
and

K E X R , , then K E ! I J ~ , + , .

Proof. Let A be any closed unbounded subset of (A,B nA) < < ( K , B), where

K ,

choose A >

< so that

B = *{!IJl,,r\lK},,<~*A.

Now (A,A nA) < < ( K , A ) , hence A E A . Also for each v] < 5 , (A,!IJl,, nA) <<(K,W,,n K ) , and each sentence “every closed unbounded subset of K intersects !IJlb” is true in (9lK,93,,n ~ since ) , KE%JI, E 9X,,+l, hence each of these sentences holds in 9?,,n I.), hence A E n,,9X,+, = ‘$I Hence ,. each closed unbounded subset of K intersects !DIE,,i.e. K E!IJ~,+ .

(a1,

,

55. The effect of structure properties. Let r be a class of subsets of R and to avoid repeating a cumbersome notation, put for this section

A =

rnlr.

e try to find lower bounds for o(A) by assuming that I‘ satisfies various structure properties, e.g. closure properties, parametrization, prewellordering, etc. THEOREM 3 . Assume A D . Let r contain all singletons and be closed under continuous substitution, &, v, 3m,V m and either 3ct or V a . Then o(A) is a cardinal. Moreover, i f is a prewellordeing in A with length t , then f o r each A c 5 , C o d ( A , is in A .

s s)

Proof. Take the case when r is closed under 3ct, the dual case following by applying this to TI‘,suppose i < K = o(A) and f:A + K is a bijection, let 5 be a prewellordering in A of length A . Since A > 0 , we can assume that Field( 5 ) = R . Consider the relations on A P(V, 5) Q(v],

* f(?) 6 f ( 5 ) 5) * f ( v ] ) < f ( 5 ) .

9

s),

By Lemma 5 both Cod(P; C o d ( Q ; 5) are in Zt(s) so that by the closure properties of I’both these sets are in r. The relation ct

s’p

e- ( c t , b ) E C o d ( P ;5 )

is evidently a prewellordering of R with length IC , since if 4 : R 3 ) A is the canonical order-preserving surjection for the prewellordering 6 , we have

s ‘ P * f(4W 5 f(4(P))-

YIANNIS N. MOSCHOVAKIS

48

r , and

Now 6 ' is in

also

1 ( a S ' PI

* f(4(B)))

* ( P , C o ~ c o d ( QS; ) , so that 5 ' is in i r , i.e. 5 ' is in A and the closure properties of imply that K -t 1 is also realized in A , which is absurd. The second part of the theorem follows easily from Lemma 5, applied to A and 5 - A . This result already shows that, assuming A D , each o(Ai) is a cardinal. In $7 we shall argue that in L [ R ] each class Zz is closed under 3m, Vm, 3a, V a ; under this hypothesis and A D , the theorem also implies that each o(Ai) is a cardinal. The remaining results of this section will imply that if we assume A D (and other hypotheses, true in L [ R ] ) , then each o(Ai) and each o(Ai) for odd k are regular. The problem is open for o(A:) with even k > 0. The dyadic second order language f o r two orderings has two sorts of individual variables, relation symbols 5 S2,=, E and six sorts of set variables over subsets of each of the individual domains and over subsets of the four Cartesian products determined by the two individual domains. For each pair 1 , of ~ ordinals, the standard model of this language is

,,

%.K

=

( 1 9 4

A

K

2, 2

(AxA)2

(AXK)2,(KXA)2,(KXK) Y

2, SA, S K4 ,

7

where S A ,S Kare the orderings on 1and K respectively. In $3 we extended the language of analysis by prime formulas of the form ~ ( 6 2): m in order to define Z ~ ( X )It. is clear how we can extend the language of analysis by relation symbols P:, .. ,P,*, each P,? suitable for denoting a subset of some product space Xi.We call formulas of this extended language analytic in P r , ...,P : . A relation definable by a formula analytic in P:, ..., P: when we give specific values P , , ..., P , to the symbols P:, ..., P: is analytic in P , , ..., P,(projective in P , , ..., P , if we also allow constants from R as parameters). Let 5 1 , S be prewellorderings of subsets of R , let 4 : Field (S + 1, t,b : Field (52) -+K be the canonical surjections. To each subset A c1 x K , put Cod(A;S

S 2)

= {(a,

p): a E Field(

and similarly for subsets of let us also put

IC x

Cod(A;S Cod(B;S

1,

,) & p E Field( S

2)

& (4(a),$(P))EA}

1. In order to have a uniform notation, jz) = C o d ( A ; 5 ,), 2)

=

Cod@; 5 J ,

y o subsets A G 1, B E IC, at least for the next lemma.

DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM

49

L E M M A8. Assume A D . To each formula 8(x,, ..., x,, y,, ..., y,) of the dyadic second order language f o r two orderings with the exhibited individual variables and set variables A,, A,, ... we can assign a formula B*(a,, ..., u,,,P1, ..., p,), analytic in the symbols 5 S 2*, A,*, A: ,... with the following property. I f S 2 , are prewellorderings of subsets of R with lengths I , K and canonical surjections 4 : F i e l d ( 5 ,)++I,$: F i e l d ( S , ) ++ K , i f B,, B,, ... are values f o r the set variables A,, A,, ... in the standard model and i f we interpret S : , S;,A:,A,*, ... b y Cod(B,; S , , g , ) , Cod(B,; 5 5,), ..., respectively, then f o r a,, ..., a,, E Field( 5 ,), p,, ...,p, E Field( 5 ,) we have

,,

,,

O*(a,,

...,anrPI,

**-,Bm)

*

3 A . K

C 6 ( 4 ( a l ) ,..-,4tan),+(Pl), * . . ? $ ( P r n ) ) .

Proof is by induction on the construction of the formulas O(x,, ...,x,, y , , ...,y,) = 8 . The assignment of 8* to 8 for prime 8 is trivial, e.g.tox, ~ , x , w e a s s i g n a , $Ta,,to y l = y, weassignp, ~ T f i &f12s;p1 , to ( x , , y , ) E A , we assign (a,, pl) E A,, etc. The induction steps involving the propositional connectives and quantifiers over I or K are also trivial, e.g. to 3xO(x) we assign 3a[a 5 ,a & 8*(a)]. It only remains to show how we deal with the set quantifiers. Given S 5 ,, let 5 be the lexicographical prewellordering on the Cartesian product of the fields.

,,

(a,p)

s ( a ‘ , p ’ ) o a , a ‘ ~ F i e l d ( S , )& P , P ’ ~ F i e l d ( s , ) & { [ a s ,a’&-(a’S ,a’)] v [ a S ,a’ &a‘ sla&p~ , p ’ ] )

The length of 5 is precisely K - I , i.e. the order type of I x K ordered by the lexicographical wellordering. If x:Field( S ) + K .I is the canonical Ix K ++ K . A the unique similarity, it is immediate that surjection and f : , the diagram Field(

s)------x

++ K . 1

commutes where (4,$)(a, p) = (4(a), $(p)). From this follows that for every subset B E I x K , C o d ( f [ B ] ;5 )

=

{(a,/3):a~Field(~,)&p~Field(S,)&(~(a),$(~))EB}

= C o d ( B ; 51, 5 2 ) .

50

YIANNIS N. MOSCHOVAKIS

,,

Thus by Lemma 5 , for each subset B c 1 x K , C o d ( B ; 5 5 ,) i s Z t ( 5 ) as a subset of R x R . Now G( 5 ) is analytic in 5 5 by a formula that we can construct from (3-3). Moreover the relation

,.

(5-1)

Q(P)

is also analytic in 5 (5-2)

0

(3B c 1 x

K)

[G,( 5 ) = Cod(B; 5 1, 5 ,)]

,,5 ,, since it is equivalent to

(VY,Y’,696’) [(P, Y,6) E G( 5 )&Y 5 17’ &Y‘ 5 1 Y & 6 5 , 6 ’ & 6‘ S26 =- (j3,~’,6‘)E G ( s ) ] .

Thus if O*(A*) is assigned to O(A), with A a variable over subsets of 1 x K, we assign to 3AO(A)

IP[Q*(P) 8~O*(Gp( 5 ))I

9

where Q*(P) is the formal version of Q ( B ) . The quantifiers over subsets of the other cross products of 1 and K or 1 and K themselves are treated similarly, so that the proof is complete. This lemma embodies many of the useful consequences of Lemma 5. It is obvious that it holds for the full second order language for any number of orderings.

THEOREM 4. Assume A D . Let A be a class of sets containing all singletons and closed under continuous substitution, &, 1,3 m , 3u. Then o(A) is a limit cardinal. Moreover, if 5 i s a prewellordering in A with length 5 , thenforeachAc(,Cod(A;s)isinA. Proof. To prove that o(A) is a limit cardinal, let A be the length of some prewellordering 5 in A , we shall show that A+ is also realized in A . In the dyadic second order language for one ordering (a sublanguage of that for two orderings) with variables A , , A,, ... varying over subsets of 1 x 1 in the standard model % A , let O(Al,A,) be the formal version of “ A , , A , are wellorderings of iland there is some subset of 1 x 1 which is a n isomorphism of A , onto A , or onto an initial segment of A,”. Let O*(A:, A:) be the formula analytic in 5 *, A,*,A,*assigned to O(A,, A,) by Lemma8, let G( 5 )be universal for the Z;( 5 )subsets of R x R . Clearly G is analytic in 5 , hence in A . As in the proof of Lemma 8, the relation Q ( E ) 0 ( 3 A C il x 1)[G,(5)

= Cod(A;5)]

is analytic in 5 , hence in A . Put then

5 ’ P * Q(a) 8~Q(P) & e*(cd5 1, C,( 5 1)-

DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM

51

Now 4‘is in A , and it is a prewellordering with length A+. The second assertion of the theorem is immediate from Lemma 5.

THEOREM 5. Assume A D . Let r be parametrized, containing all singletons and closed under continuous substitution, &, V, 3 m , ’dm, 3u and V u . T h e n o(A) is a regular limit cardinal (inaccessible). Moreover, if 5 is a prewellordering in A with length 5 and A E 5 , then C o d ( A ; 6 ) is in A .

Proof. Let K = o(A), the preceding theorem implies that K is a limit cardinal and the second assertion of the theorem follows from Lemma 5. It remains to show that K is regular. Let 1 be the length of some prewellordering 6 of R in A and assume that f:1 + K is a cofinal map - we shall obtain a contradiction. We may assume that for each q , f(q) > 0 . Let G be universal for the subsets of R x R in r , put g(q) = {(Ply):

G, = R x R - G , and G , is a prewellordering of R with length f(q)}

.

By hypothesis g(q) # Pr for each q , so by Lemma 3 there is a choice subfunction g* of g such that Cod(g*, is in Zt(s), hence by the closure properties of A , Cod(g*, EA. If (a, p, y) E Cod(g*; 5 ), then G is a prewellordering of R with length f ( ~ ( L x ) )where , 4 : R + I is the canonical order preserving surjection for 6 ; let 4,: R + f (4(a)) be the canonical order preserving surjection for G,. Put

s)

s)

(u,P,y,b)

5 ’ (a’,P,y’.6’) * (u,B,Y)EcOd(G*; 6 ) & (a’, p ’ , y ’ )

4/?(6)6

E Cod(g*;

$0,

6)

(6’).

Now 5‘ is a prewellordering of a subset of R4 with length precisely the canonical surjection being $(u, P, 7, 6)) =

K,

$/?(4 *

We shall obtain a contradiction and complete the proof if we can show that 5 ’ is in A . In the dyadic second order language for two orderings, with x, x‘, ... varying over the first ordering and y , y‘, ... over the second, let $(x, y ) be the formula which defines the relation x 6 y , i.e. there exists a subset of the cross product of the two domains which is an order preserving bijection of { x ’ : x ’ 6 , x } onto { y r : y ‘ 6 z y } . Let B * ( a , p , j E 5;) be the

52

YIANNIS N. MOSCHOVAKIS

analytic in ST,5 ; formula associated with O ( x , y ) by Lemma 8. If (a,p, y, S), (a’, p‘, y ’ , 6’) are in the field of 5’ , then Lemma 8 asserts precisely that

4p(@ 5

4p*(S’)

* 0*(4S‘,

s;,s;)

whenever 5 ; interprets the relation G, and S 2 the relation Gpt. But G, = R x R - G,, G i = R x R - G y p ;if. we substitute G,(G,.) in all the positive occurrences of 5 I” (5;)and R x R - G, ( R x R - G y p )in all the negative occurrences of 5 ; (ST), then the closure properties of r imply that the resulting relation is in r . The dual substitution gives us a relation in i r ,and both these relations are equivalent to 4p(S)5 q5a.(S’) whenever (a,p, y , S), (a’, p’, y’, 6’) are in the field of 5 ’, i.e. (a, p, y), (alp’, y ’ ) ECod(g*;5 ) . Thus 5 ’ is in A and the proof that K is regular is complete. If we want to eliminate the hypothesis that r is closed under both 3a and V a , we must add the prewellordering property to the assumptions. THEOREM 6 . Assume A D . Let be parametrized, containing all singletons and closed under continuous substitution, &, v, 3m, Va, assume Prewellordering (r).Then o(A) is regular. Moreover, if 5 is a prewellordering of the universal set G with length K which establishes the prewellordering property, then K = o(A) and f o r each A c K , C o d ( A ; is in r .

s)

Proof. The statement of the prewellordering property implies immediately that K 5 o(A), since for each 1< K , if 4 : G + K is the canonical surjection and 4(xo) = A , then { ( y , z ) :y

(5-3)

5

z & z j xg &

l(Xg

5

z)}

is in A and has length A . Assume 5’ is a prewellordering in A with length I and canonical surjection rl/: Field( 5 ) + 1 , assume that there is some f:A + K such that supremum{f(r]):r] < A} = K . We shall obtain a contradiction from these assumptions, thereby showing at the same time that o(A) 5 K and that K is regular. Put

f *(?I

= {x: x E G

4(x> = f ( r ] ) }

and by Lemma 3 let g* be a choice subfunction o f f * with C o d ( g * ; I ‘) in Xi( 5 ‘). The closure properties of r imply that C o d ( g * ; 5 ’) is in ir . Now ~ E -G s 3 a 3 x [ ( a , x ) ~ C o d ( g *5‘) ; &y

where

5

is in

i r and agrees

with

5 x],

5 when its second argument is in G

DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM

53

as in (2-2), which implies G E i r ,violating the remark at the end of 2.1 (the “hierarchy theorem”). It remains to show that for each A E K , Cod(A; 5) is in r - notice that this does not follow immediately from Lemma 5 , since r is not assumed closed under 3a. For each 1< K , let be the initial segment of 5 up to and not including points that correspond to 1, defined by (5-3). For each A c 1, Cod(A; is in Xt(SA),hence in i r by the closure properties of r; but this implies that for each A E 1, Cod(A; S A )is in r, since

sA

sA)

Let H be universal for the subsets of R x R in r, let n : R + R x R be the canonical homeomorphism, let A E K , consider the following twoperson game. If I plays a and I1 plays p ,

I1 wins

-

7ca $ G v [nu E G &

(31 >-+(a))[ H , = Cod(A (7 1;&)I.

(Here of course 4: G ++K is the canonical surjection.) Suppose I wins the game with a strategy o ,we shall obtain a contradiction. We must haveforallp, n(o* [ ~ ] ) ~ G a n d s u p r e r n u r n { + ( n ( a * [ ~ ] ) ) : ~ ~ R } = ~ , since if for all p , q5(n(o*[PI)) < 1, then I1 could win by playing p so that H , = Cod(A n1;&). Therefore we have

which implies that G E i r , a contradiction. Now I1 must win, by some strategy t , and we have xECod(A;

s ) * XEH[,-l,,,r

which shows that Cod(A;x ) E , completing the proof. If we assume that r satisfies the closure properties of Theorem 5 uniformly, in a sense we shall explain shortly, then we can prove K = o(A) without assuming A D . This result (and the lemmas leading to it) has independent interest in hierarchy theory and is the best characterization of o(A) that we know, without assuming axioms that contradict choice. For these reasons we give it here, even though it is not directly related to A D . If r is parametrized by G , then r = {Ga:ct E R } and for each A E r, we call {a:G , = A } the set of codes for A . In (2-1) we assigned to each parametrization of r a canonical parametrization of [X]r (when r is closed under recursive substitution) so that we can talk about codes for

54

YIANNIS N. MOSCHOVAKIS

A ~r when A E X, for any product space X. Our standard hypotheses up till now are that r is closed under various operations, e.g. continuous substitution, &, v , etc. Let us say that r is uniformly closed under an operation if the result of applying that operation is always in r and a code of it can be computed from codes of the arguments via a continuous function. To be precise for some of the operations that we shall be most concerned with: r is uniformly closed under continuous preimages if there exists a continuous f:R x R + R such that if { E } ~ is ; ~ completely defined, then Gf(,,,)= { p : { E } (p) E G,} . r is uniformly closed under &, if there is a continuous f :R x R + R such that for each x , P , G,,,,,, = G, n G,, and similarly for v . r is uniformly closed under 3m if there is a continuous function f:R + R such that for each a , G,,,) = { p : 3m(m, p) E H a } , where H is the canonical universal set for [w x R ] T , defined from G via (2-1), and similarly for V m , 3a, Vcr.

LEMMA 9. Let r be parametrized b y G , containing all singletons and uniformly closed under continuous substitution, &, v , 3m, V m , Va, assume Prewellordering (r)via a canonical surjection 4 : G j) K . There i s a continuous function f l :R x R + R such that i f A = R - G, and i f f o r each p E A , { E } (p) E G , then f ( E , cr) E G and (Vp E A ) [ 4 ( { ~(p)) } <

< 4 u( E , 4 1 1 *

The import of the lemma is that if a continuous partial function is defined over some A E l l - and takes values in G , then the set of these values is bounded in the prewellordering. We need the uniform version of this lemma, however, in the way that we stated it.

Proof. Put

* 3Y[YEA

{ E I W is

&(P,P>

5 {E}(Y)I

where $ is in il-and agrees with S when its second argument is in G , as in (2-2). Now the closure properties of r imply that B E i r ;moreover, because of the uniformity hypothesis, there is a continuous g , :R x R + R such that if A = R - G , , then B = R - G g , ( E , ai.e. ), Put and assume A = R - G , and that for each p E A , { E } (p) E G . If f,(E, a) .$ G, then by definition gl(E,cr) E B , hence for some y E A , (gl(E,a), g l ( q a)) = fl(c, M) 5 { E } (y), which implies fl(e, cr) E G , contradiction; hence S ~ ( E , M ) E GBut . then gl(E, a ) $ B , which implies that for each y E A , l(fl(E, a) 5. E l (Y), i.e. 1(fl(-% a> (4 (Y), i.e. 4 ( { E } (7)) < 4(fl(&, a>>.

s

DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM

55

LEMMA10. Let r be parametrized b y G , containing all singletons and uniformly closed under continuous substitution, &, v , 3 m , V m , V a , assume Prewellordering (r)via a prewellordering S on G with length IC and canonical surjection 4 : G ++IC.There is a continuous function f z ( a ) such that if H is the universal set f o r [ R x R ] T associated with G via (2-1) and 5’ = H a = R x R - H , is a prewellordering with length t , then fz(.,P>EG and t S +(fz(a,P>>. Again, the import of the lemma is that every prewellordering in A has length less than K , but it is this stronger, uniform version that our proof is suited for. Proof. Using the uniform closure properties of r we easily show that for some continuous gl(a,/3,y), if 5’ = H a = R x R - H , is some relation in A , and y 5’ y is some element of its field, then R - Gg,(a,p,y) = (6: 6 2 ’ 7 & -I(Y

S’S)}.

Using the recursion theorem for continuous partial functions, choose so that (5-4)

{E*I(%P,Y)

2:

E*

f,(g,(x,P,Y),SRxR(&*,4 P > > ,

where f , is the function of the preceding lemma and S R x Ris the recursive function of the iteration theorem, so that (5-5)

{E*kP?Y)

=

{S

RXR

(E

*, % P ) } ( Y ) .

Notice that { E * } (a, P, y) is a completely defined function, since fl and g , are completely defined. Assume that S ’ = H a = R x R - H, is a prewellordering with length t and canonical surjection I):Field( 5 ’) + t . We prove by transfinite induction on I)(y) that if y E Field( ’), then {&*}(a, P, y) E G and $(y) S + ( { E * } (a, P, y ) ) : assuming this for all 6 with $(S) < $ ( y ) , the preceding lemma, (5-4) and (5-5) give us immediately that for all 6 with $(a) < $(YL we have 4 ( { E * } (a, P, 6 ) ) < 4({E*}(rn,P,Y)), i.e. I)@) < < 4({&*} (a, P, Y)) hence I)(r) S 4({&*} (a, P, Y)) * The lemma now follows if we choose by the uniform closure a continuous g , : R x R + R such that if 5‘ = H a = R x R - H B , then 9

and put

R-

fz(% PI

GgL(a,p)

=

{Y:Y S ’ Y }

= f i ( g Z ( 4 P), S R X (&*, R a, PI) *

THEOREM 7. Let r be parametrized b y G , containing all singletons and uniformly closed under continuous substitution, &, v , 3 m , Vm, Va,

YIANNIS N. MOSCHOVAKIS

56

assume Prewellordering T h e n K = o(A).

(r)via

a prewellordering 5 on G with length

K.

Proof. That K S o(A) follows from the statement of the prewellordering property, as in the proof of Theorem 5. That o(A) 5 K follows immediately from the preceding lemma. 56. Projective ordinals. For each k > 0 , let

6: = o(A:),

We collect in one theorem the results about these ordinals which follow from what we have already shown. THEOREM 8. (8.1) Assume Determinacy ( A i m ) , DC. Let 5 be a prewellordering of of a universal lT:,+, set which has length K and establishes Prewell1 ordering (ll;,,,+,); then K = 62m+l. (8.2) Assume A D . (8.2.1) Each 6: is a cardinal. (8.2.2) If 5 is a projective prewellordering with length 5 and A G 5, then C o d ( A ; 5 ) is a projective set, hence A is dejinable ( i n set theory) f r o m a real parameter. (8.2.3) I f 5 < 6; and A = with each A , , E E ~ ,then A E E : . (8.2.4) A set A is in El if and only if it is the union of Kl Borel sets. (8.3) Assume A D , D C . (8.3.1) Each 6im+l is a regular cardinal. (8.3.2) If 5 is a prewellordering of a universal IIfm+, set which has length 6f,+l and establishes Prewellordering (llf,+l), then f o r each A c 6i,,,+,, C o d ( A ; 5 ) is II;,+,. (For m = 0 this is due to Solouay.) (8.3.3) For each m 2 1, 6;,,,-, < 6f, 5 6:,+,. (8.3.4) For each m Z 1, (6:,< 6:,+, . (8.3.5) For each k, K: 5 6:.

U,,<&,

Proof. (8.1) follows from Theorem 7 and the Prewellordering Theorem, (IV), of $2. (8.2.1), (8.2.2) follow from Theorem 3, (8.2.3) from Lemma 6 and (8.2.4) from (8.2.3) and the classical result that every El set is the union of K, Borel sets. (8.3.1) and (8.3.2) follow from Theorem 6 and (IV) of $2. The firstChalf of (8.3.3) follows from Theorem 6 and (IV) of $2 (which imply that 6:,,,-1 is the length of a ll:m-l, hence A:, prewellordering) and the second half is trivial. (8.3.5) follows immediately from (8.3.3) and (8.3.4) so that we need only give a

DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM

57

Proof of (8.3.4). Let SZrn-' be a prewellordering on a set G Z m - ' , 1 universal for nzm, which establishes Prewellordering (IIi,- 1) by (IV) of $2. By Theorem 6 , each subset A of has Cod(A ; 5 " " - l ) in II;,,,- 1 , and the same holds by a trivial extension for every binary relation A on 62,- 1 . If H Z m - l is universal for the sets of pairs in l-Ii,,,-l, it is now trivial to verify that the set a E WZrn-' o H:"'-'

= Cod(A; 5)f o r some wellordering A of 6iml

is in Hi,,,. Let G2"'+' be universal for n:,,,,,, let +2m+1:G2m+1 + 6:m+1 be the canonical surjection for a prewellordering that establishes prewellordering (IIj,,,+ l). Using the trivial observation that each I: , set is "uniformly" in A i m + and Lemma 10, we easily obtain a continuous function f ( a ) such that a E W Z m - ' + f ( a ) E GZm+l

and if H:m-l = C o d ( A ; S Z m - ' for ) some wellordering A with length tA, then tAS +'"'+'(f(a)>. Now Lemma 9 implies that for some q < 1 + and all such A , S q , so that (6zm-l) S q .c S:,+l It is a classical result that 6; = 6: = K1.In view of (8.3.5) above one would hope that with A D and D C either 6: = K: for all k 2 1 or 1 1 = K:,+, for all m . 6,, = It would be nice to be able to prove from A D and DC that every 8: is regular. However we only know this for odd k and it is not clear that our methods can be extended easily to prove it for even k. Solovay has proved that K, and K, are measurable assuming A D . It should be the case that with A D and D C all 61m+ are measurable, perhaps also all ti;,,,. (It can be easily shown that each 6&+l carries a countably additive complete measure such that the measure of every bounded set is 0 .)

.

$7. A:.

For each k 2 1, put 6," = o(A,").

The key to obtaining results about 6: with our methods is to prove closure properties about the classes Et,I , A t . It is easy to prove from the de: finitions, Lemma 1 and trivial contractions of variables that Z: II: are parametrized and closed under continuous substitution, & and v . However the usual proofs that these classes are also closed under 3m Vm,3a, Va use the following special case of the axiom of choice. For each F : R + w put FJB) = W ( a , B)),

YIANNIS N. MOSCHOVAKIS

58

where z: R x R + R is the canonical recursive homeomorphism. The axiom of choice then implies that for each A E R x R u (7-1)

Va3F(cr,F ) E A

* 3FVa(a, F,) E A

and (7-1) then implies easily that Ei, nz are closed under 3m, V m , la,Va. We do not know if (7-1) is consistent with A D ; but the same closure properties can be proved from the more innocuous collection property, which we can establish at least in L [ R ] , Collection. For each A

G

R x

Va3F(a, F ) E A

"0,

*

3FVa3/?(a,F,)

E

A.

LEMMA11. Let M be a class which is a model of Z F (without choice), which contains R and such that some surjection f : O N x R -+M is definable in M with parameters f r o m M ; then M satisfies Collection. I n particular L[R] and the class of sets hereditarily ordinal definable f r o m real numbers satisfy Collection. Proof. Assume V

=

M and Va3F(a, F ) E A , for some A

v(a) = infimum

G(% 79 6) =

{t:3y[f(t,r)

f ( v ( 4 , Y) (6) if f

= 0

E Rm &

cR

x Ru.Put

(a,f(t,r)) E 4 1 ,

Y) E Ru,

otherwise,

and finally

F(&) = G(a,y,G) for the unique a, /?,y , 6 such that n(/?,6) = E and z(a,y) = 8 . Now for each a , choose y so that f ( ~ ( a )y ,) E Rw and (a,f ( ~ ( a )7)) , E A ; if /? = n(a,y ) , then by the definition F,(@ = F(n(/?,6)) =

i.e. F,

= f ( ~ ( a )y),

, hence (a, F,)

E

f Ma), ?)(a 7

A.

THEOREM 9. Assume A D and Collection. Then each 6: i s a regular limit cardinal (inaccessible). Moreover, if 5 is a A; prewellordering with length 5 and A 5 5 , then Cod(A; 6 ) is A:, hence A is definable (in set theory) f r o m a real parameter.+ f h o t h e r very easy consequence of Collection is that o ( R 2 ) is a regular ordinal. Thus with AD and Collection, o ( R 2 ) has fairly strong reflection properties, since it is e.g. a regular limit of Mahlo cardinals. (We do not know how to prove with AD and Collection that o ( R 2 ) is Mahlo.) Solovay proved first that o ( R 2 ) is regular in L [ R ] -this follows from our Lemma 11. We doubt that the regularity of o ( R 2 ) can be shown without any choicelike axioms like Collection.

DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM

59

Proof is immediate from Theorem 5.

Using the hypotheses of Theorem 9, it is not hard to see that 6: is a limit of inaccessibles, a limit of regular limits of inaccessibles, etc. However, we can get much better lower bounds for 6: if we also assume DC.

THEOREM 10. Assume AC, DC, Collection. Let L be the dyadic second order language f o r an ordering,for each 1 let 2, be the standard model of L as in $4. Then thefirst regular K which rejects f o r this L, is less than 612.7

{aA}

Proof. Let Put

71: R

F E WF

x R

0

-,R

be the canonical recursive homeomorphism,

{(a,P): F(n(cl

p))

=

O } is a prewellordering.

Now DC implies immediately that F E W F is an analytic relation, i.e. it is defined by a formula of third order number theory without quantifiers over Rm. If F E W F , let S F be the prewellordering determined by F , let $ F : F i e l d ( 5 , ) -+ 1 , be the canonical surjection onto the length AF. Let K be the least ordinal that reflects with respect to L, {a,}, let $ : ~ + ~ 2 be the mapping defined in the proof of Theorem 2 which assigns codes to all ordinals less than K . Put Q(F)

(7-2)

R(F, G)

(7-3)

0

FEWF&& =

K ,

* F E W F & VclVp [m E Field ( 5F) => [G(n@,P)> = 0 * P E $(*F(4)11*

We wish to show that both Q(F) and R(F, G) are A; , i.e. they are definable both by Z; and l-I? formulas of third order number theory. If we can do this we will have proved the theorem, since then the prewellordering S K of the proof of Theorem 2 will be given by ~1

sKB*

3F3G[R(F, G) & 3 y 3 6 [ F ( ~ ( y , 6 ) )= 0 & G(n(y,CO) = 0

0

G(n(6,B)) = 011

V F V G [ Q ( F ) & R(F, G) * 3y36[F(n(y,6)) = 0 & G(n(7,a)) = 0 & G(x(6,P)) = 0 ] ] .

We shall outline the computation of Q ( F ) , R(F,G) in a sequence of sublemmas. t This result was obtained after the Jerusalem meeting. At the same time Solovay also proved independently and by a different method that (with A D ) there are highly Mahlo cardinals less than 6:.

YIANNIS N. MOSCHOVAKIS

60

SUBLEMMA 10.1.

Put

P , ( n , m , F , G , ) 0 [ n is the Godel number of a formula O(S*,A*,uI, . . . , u r n )analytic in the binary symbol and the unary symbol A* and with m f r e e variables] & [e( S *, A * , ( U ) .~. ., , (cl),) is true when we interpret 5* b y S F and A* b y { P : G ( p ) = O } ] .

s*

T h e n Pl(n,m,F,G,cc) is a At relation. Proof of this sublemma is by the usual analysis of the induction involved in the definition of truth and we shall omit it.

SUBLEMMA 10.2. Let H ( F ) be the canonical universal set f o r C: ( S F ) , PU t P 2 ( F , f i ) 0 f o r some B c I F , H p ( F ) = Cod(B; SF). T h e n P,(F,P) i s analytic. Proof. H ( F ) is analytically definable from F and then

P,(F, p)

0

vcc[u E Hp(F) => CL E Field ( S ,)] & V c t V y [ c c ~ H ~ ( F ) &~ a F &Y y

YEH~(F)].

SUBLEMMA 10.3. Put P 3 ( n , m , F fi,cc) B(x,,

0

[ n is the Godel number of a formula

...,x,, A ) of the dyadic second order language

f o r one ordering with m f r e e individual variables and one f r e e set variable] & F E W F & Vi[l & [‘u,, k O ( $ F ( ( ~ ) , ) ,

S i5m

( ~ ~ ) ~ s F i e( SZFd) ] & P 2 ( F , P )

...,~ j ~ ( ( x ) , ) , B ) ,when

B is such that

ffp(F) = C o d ( B ; S F ) ] .

T h e n P,(n, m, F , f i , a ) i s A;. Proof is immediate from Sublemmas 10.1, 10.2 and Lemma 8. SUBLEMMA 10.4. Put

P,(F,

p, y )

&

( 5F) & P,(F, 8) [if B G A, is such that H&F)

=

C o d ( B ; &),

o F E W F & y E Field

then Then P,(F,P, u ) is A;.

($F(y)

7

$F(Y))

(lF,

B)l *

DETRMINACY A N D PREWELLORDERINGS OF THE CONTINUUM

61

Proof. The last condition in P,(F,P,y) is equivalent to

VnVmVa{[Vi[l S i 5 m

=>

[(a)i S F y & l Y s F ( a ) i ] ] & P3(n,m,FY,flY,tI)]

=>

Pdn, m, F , P, a>>

7

sFv sF),

where FY,PYare chosen so that is the restriction of S F to the points less than y and if H,(F) = Cod(B; then H,y(FY)

=

Cod(B n $F(y);

5FY) .

SUBLEMMA 10.5. The condition Q ( F ) defined b y (7-2) is A:. Proof. Using the preceding sublemmas, put

P,(F) o F

E

W F & ,IF is regular & VflVY[P,(F, P ) & y

3Y“Y

SFY’

E Field

(s

F)

&P,(F,P,Y’)ll.

Notice that the condition “AF is regular” is A:, since it is equivalent to P3(no,B,F , Po, ao) for some fixed n , , p , , a , , so that P,(F) is A t . Now Q(F) o P , ( F ) & V y [ y E Field (&)

1P5(FY)].

The mapping 4: K -+ R2 was defined by transfinite induction on 5 < K , where for each 5 < K there were four cases. In order to prove that R ( F , G ) is A;, we must show that each of the case hypotheses, and then the definition in each case are A: (in terms of the coding of ordinals provided by F ) . One can write down all the clauses quite easily, by applying the sublemmas above, but it is a tedious mess and we shall avoid committing it to print. This proof of Theorem 10 can be directly relativized to any given prewellordering in A: so that it yields the following result: f o r each 5 < S:, there i s some K , 5 5 K < S:, such that K reflects with respect to L, {aA). In particular, 6: is a limit of ordinals K which are in mK. Perhaps we should remark that Theorem 9 is easily extended to the classes A:, with n > 2 . It is easy to formulate the appropriate collection :, Ill:are closed under 3m,Vm, la,Va property which allows us to prove that E and then show that it holds in L [ R ] ; this implies then that Sl: = o(Ai) (n 2 2, k 2 1) is a regular cardinal. REFERENCES Separation principles in the hierarchies of classical and effective [l] J. W. ADDISON, descriptive set theory, Fund. Math., vol. 46 (1959a), pp. 123-135. [21 , Some consequences of the axiom of constrirctibility, Fund. Math. 46 (1959b), pp. 337-357.

62

YIANNIS N. MOSCHOVAKIS

[3] J. W. ADDISON and YIANNIS N. MOSCHOVAKIS, Some consequences of the axiom of definable determinateness, Proc. Nat. Acad. Sc. USA 59 (1968), pp. 708-712. Infinite games with perfect information, Ann. Math. 141 D. GALEand F. M. STEWART, Studies 28 (153), pp. 245-266. [5] K. GODEL,The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Ann. Math. Studies, No. 3, Princeton University Press, Princeton, N. J. 1940. Introduction to metamathematics, Van Nostrand, Princeton, N. J. 161 S. C. KLEENE, 1952. 171 , Countable functionals, Constructivity in mathematics, North-Holland, Amsterdam 1955. [8] D. A. MARTIN, The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin Amer. Math. SOC.74 (1968), pp. 687-689. 191 Y. N. MOSCHOVAKIS, Hyperanalytic predicates, Trans. Amer. Math. SOC. 129 (1967), pp. 249-282. [lo] --, Abstract first order computability. 11, Trans. Amer. Math. SOC.138 (1969), pp. 465-504. On the axiom of determinateness, Fund. Math. 53 (1964), pp. 205-224. [ l l ] J. MYCIELSKI, JR., Theory of recursive functions and effective computability, McGraw1121 H. ROGERS Hill, New York, 1967. A complete classification of the A; functions, Bull. Amer. Math. SOC. [13] Y.SUZUKI, 70 (1964), pp. 246-253.