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# The Main Results

Our context is the study of -models of subsystems of second order arithmetic [5, Chapter VIII]. As in [5, Chapter VII], a -model is an -model M such that, for all sentences with parameters from M, is true if and only if . Theorems 1.1 and 1.3 below are an interesting supplement to the results on -models which have been presented in Simpson [5, §§VII.2 and VIII.6].

Let HYP denote the set of hyperarithmetical reals. It is well known that, for any -model M, HYP is properly included in M, and each is definable in M.

Theorem 1.1   There exists a countable -model satisfying
if X is definable, then .

Proof.Fix a recursive enumeration Se, , of the sets of reals. If p is a finite subset of , say that meets p if for all . Let be the set of p such that there exists meeting p. Put if and only if . Say that is dense if for all there exists such that . Say that is definable if it is definable over the -model HYP, i.e., arithmetical in the complete subset of . Say that is generic if for every dense definable there exists such that meets p. We can show that for every there exists a generic meeting p. (This is a fusion argument, a la Gandy forcing.) Clearly is a -model. We can also show that, if C is countable and , then there exists a generic such that . Let be the language of second order arithmetic with additional set constants Xn, . Let be a sentence of . We say that p forces , written , if for all generic meeting p, the -model satisfies . It can be shown that, for all generic , the -model satisfies if and only if meets some p such that . If is a permutation of , define an action of on and by and . It is straightforward to show that if and only if . The support of is defined by . Clearly if and , then . We claim that if and are generic, then the -models and satisfy the same L2-sentences. Suppose not. Then for some we have and , for some L2-sentence . Let be a permutation of such that . Since , we have , hence , a contradiction. This proves our claim. Finally, let where is generic. Suppose is definable in M. Let be generic such that has . Let be an L2-formula with X as its only free variable, such that exactly one , and . Then exactly one . Let be such that . Then for each , we have that if and only if and , if and only if and , if and only if . Thus A=A'. Hence . This completes the proof.

Remark 1.2   Theorem 1.1 has been announced without proof by Friedman [3, Theorem 4.3]. Until now, a proof of Theorem 1.1 has not been available.

We now improve Theorem 1.1 as follows.

Let denote hyperarithmetical reducibility, i.e., if and only if X is hyperarithmetical in Y.

Theorem 1.3   There exists a countable -model satisfying
if X is definable from Y, then .

The -model which we shall use to prove Theorem 1.3 is the same as for Theorem 1.1, namely where is generic. In order to see that M has the desired property, we first relativize the proof of Theorem 1.1, as follows. Given Y, let be the set of such that there exists meeting p with X0=Y. (Obviously 0 plays no special role here.) Say that is generic over Y if, for every dense definable from Y over , there exists such that meets p.

Lemma 1.4   If is generic over Y, then G0=Y, and is a -model satisfying if X is definable from Y, then .

Proof.The proof of this lemma is a straightforward relativization to Y of the proof of Theorem 1.1.

Consequently, in order to prove Theorem 1.3, it suffices to prove the following lemma.

Lemma 1.5   If is generic, then is generic over G0.

Proof.It suffices to show that, for all p forcing is dense in , there exists forcing and meets r). Assume is dense in . Since , it follows that and . Fix and such that . Put meets . Then S' is a set, so let be such that S'=Se. Claim 1: . If not, let be such that . Let be a permutation such that and . Then and , a contradiction. This proves Claim 1. Claim 2: . To see this, let be generic meeting . By Claim 1 we have . Hence , i.e., there exists meeting q' with X0=G'0. Thus meets . This proves Claim 2. Finally, put . Then and and meets q'). This proves our lemma.

The proof of Theorem 1.3 is immediate from Lemmas 1.4 and 1.5.

Next: Conservation Results Up: A Symmetric -Model Previous: A Symmetric -Model
Stephen G Simpson
2000-05-23