Turing determinacy and the continuum hypothesis

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Arch. Math. Logic (1989) 28:149-154

Mathematical Logic ©Springer-Verlag 1989

Turing Determinacy and the Continuum Hypothesis Ramez L. Sami Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt and U.F.R. de Mathematiques, Universite Paris VII, F-75251 Paris Cedex 05, France

Abstract. From the hypothesis that all Turing closed games are determined we prove: (1) the Continuum Hypothesis and (2) every subset of �1 is con­ structible from a real.

0. Preliminaries

Given A�w"'=1R, an infinite two-person game of perfect information G(A), is defined in the usual fashion: players I and II produce natural integers in turn. After infinitely many moves, a real a = ( a ; l i (w ) E 1R is generated, as illustrated by the diagram: I

II

Player I wins this run of the game if a EA. II wins otherwise. One says that G(A) (or just A) is determined if one of the two players has a winning strategy in the game. (See Moschovakis' [Mo, Chap. 6], for complete definitions.) Let ;;=; rand = r denote, respectively, Turing reducibility and Turing equiva­ lence. A set is called Turing closed if it is saturated for = r· There is a natural one­ one correspondence between Turing closed subsets of 1R and subsets of f!fi the set of Turing degrees. We will identify the two types of sets without explicit reference. For r � 9(1R), Det(r) is the statement that all sets A E rare determined, while Turing-Det(r) is the statement that for all A E r, if A is Turing closed, then A is determined. Det(81'(1R)) is known as the Axiom of determinacy: AD, while Turing­ Det(.'?fl(1R)) is the Axiom of Turing degree determinacy or Turing AD. This last axiom is best understood in light of the following easy yet powerful lemma due to Martin (see [Je, 43.6], for a proof).

Suppose A �1R is Turing-closed and A is determined, there is a real a such that {PIJJ;;,:;ra}�A or {PIJJ;;,:;ra}nA=Ql.

Lemma.

150

R. L. Sami a

Let ocE1R. By the cone with vertex oc we mean the set {/31 f3 � rOC}, denoted by V, a similar convention holds for Turing degrees. It is easy to check that the converse of Martin's lemma is true. Thus for a class r, Turing-Det(r) is equivalent to the statement: whenever AEr is a set of Turing degrees then either A contains a cone of degrees or A is disjoint from one. After the first investigations of AD [My], the following consequences of AD were derived: (a) The Continuum Hypothesis. (b) Every subset of �1 is constructible from a real. (c) �1 is a measurable cardinal. (a) is due to Davis [Da]. (b) and (c) were proved by Solovay. Martin found a proof of (c) using only Turing AD. (See [Je, Theorem 103 and 43.7].) The results of this paper can now be stated in this perspective. In Theorem 1.3 we show that Turing AD implies the Continuum Hypothesis. In Theorem 1.4 we prove from Turing AD that every subset of �1 is constructible from a real. Our underlying theory is ZF +DC (the axiom of dependent choices). Jech's [Je] is a convenient reference for set theoretic notions and results. 1. Let 1P� denote the usual notion of forcing for collapsing the ordinal �. 1P2 is just the forcing notion for adjoining a Cohen generic real. Let � be countable and ocE1R. Assuming Turing AD, since �1 is measurable, &>(1P�)nL[oc] is countable, hence Pcgeneric over L[oc] subsets of 1P� exist in profusion. 1.1. We shall need the following construction due to Solovay and described in his paper [So, III-1.10 and Ill-1.11]: Let � the

Continuum Hypothesis.

To see this, one need only observe that the Schroder-Bernstein theorem is a theorem of ZF. We shall need the following notation. Let 1: w x w�w be a recursive bi­ jection. Given f:w�e, the code of f, noted J is a member of 2"' defined by: f(l(n,m))=1 ¢> f(n) < f(m). Observe that, for any real oc, L[ocEB]]�L[oc][f] and that if f is onto then f can be easily recovered from J, hence L[ocEB]] =L[oc][f], in this case. Proof of 1.3. Assume Turing following set of reals:

AD

and let A�IR be uncountable. Define the

S= {/31 (3ocEA)(ocEL[f3] and ocj;h/3)}. We claim that S contains reals of arbitrarily large Turing degrees. Let yEIR, since A is uncountable, we can pick aEA such that aj;h'l'· By Lemma 1.1, we can find f3 � r'l' such that aEL[/3] and aj;hf3· This shows {3ES and establishes our claim. p

Now let {3 have V � S. Set v =��[Pl. Now given g:w�v since f381gES, we can find ocEA, ocEL[f3EBg], such that ocj;h{381g. Let oc9 be the first such real in the canonical well-ordering of L[f3EBg]. If, further, g is surjective then it is easy to see that v�w1 �w1e111. Hence every real in L.[f3Ei1g] is hyperarithmetic in f381g, a fortiori so is every member of L[/3]niR. Consequently, oc9 must belong to L[/3] [g]- L[/3]. Let H:2"'�IP. be a function with properties (i) to (iii) as in 1.1 and set: F(b)

=

oc0H,

for

O* exists," [Ha]. The main question so far unsettled in this particular domain can be roughly put this way: is it true that for any "reasonable" pointclass r we have: Turing­ Det(r) => Det(r)? In particular is it the case that: Turing AD implies AD? H. Woodin has shown this to be true in the inner model L[1R]. His (unpublished) proof uses a "fine structure" analysis of L[1R]. The only other results known to us here are:

(1)

ZFf-Turing-Det(IID => Det(IID.

This follows immediately by combining two theorems of Martin and Harrington, respectively: 0"" exists => Det(JID, [Ma 1], and Turing-Det(IID => O* exists. (2)

zF- + DCf-Turing-Det(LID => Det(LID.

This is due to Friedman (unpublished). Roughly, the proof uses the techniques of [Fr 2] to get well founded models of ZF- with enough cardinals. One then invokes Martin's proof of Borel determinacy [Ma 2] inside these models. It seems plausible to conjecture that: Turing-Det(l:'�) => Det(l:'�) ,

for

n�2 .

One should observe that our proof of Theorem 1.3 has a"local" character and can be seen to use only Turing-Det(l:'�), whenever the set Aunder consideration is l:'� (n�2). The status of Turing AD in connection with AD seems more obscure to us, it should be possible however to derive "most" interesting consequences of AD from Turing AD.

154

R. L. Sami

Acknowledgement. The results of this paper are from the author's Ph.D. dissertation at the

University of California, Berkeley. It is a pleasure to acknowledge here Robert Solovay's inspiring supervision of this dissertation.

References

Davis, M.: Infinite games of perfect information. Ann. Math. Stud. 52, 85- 10 1 ( 1964) Friedman, H.: Determinateness in the low projective hierarchy. Fundam. Math. 77, 79-95 ( 1971) [Fr2] Friedman, H.: Higher set theory and mathematical practice. Ann. Math. Logic 2, 326-357 ( 197 1) Harrington, L.: Analytic determinacy and O*. J. Symb. Logic 43, 685-693 (1978) [Ha] Jech, T.: Set theory. New York: Academic Press 1978 [Je] [Mat] Martin, D.: Measurable cardinals and analytic games. Fundam. Math. 66, 287-29 1 ( 1970) [Ma2] Martin, D.: Borel determinacy. Ann. Math. II Ser. 102, 363-371 ( 1975) [Mo] Moschovakis, Y.: Descriptive set theory. Amsterdam: North-Holland 1980 [My] Mycielsky, J.: On the axiom of determinateness. Fundam. Math. 53, 205-224 ( 1964) [Sa] Sacks, G.: Higher recursion theory. Berlin Heidelberg New York Tokyo: Springer (in press) Solovay, R.: A model of set theory in which every set is Lebesgue measurable. Ann. Math. [ So] II Ser. 92, 1-56 (1970)

[Da] [Fr 1]

Received April 26, 1988/in revised form February 24, 1989