A characterization of ${\mathrm{\Sigma }}_1^1$-reflecting ordinals

Abstract

We consider clopen game formulae, analogous to the open game formulae widely studied in admissible recursion theory. This leads to characterizing the locally countable ${\mathrm{\Sigma }}_1^1$-reflecting ordinals α as those for which, over $L_{\alpha }$, the clopen-game–definable relations are not precisely those which are both open-game and closed-game definable.

Introduction

An admissible ordinal α is ${\mathrm{\Sigma }}_1^1$-reflecting if for every $\gamma <\alpha$ and every ${\mathrm{\Sigma }}_1^1$ formula (in the language of set theory), we have

$$L_{\alpha }\models \varphi (\gamma )\text{ implies }\exists \beta <\alpha (\gamma <\beta \wedge L_{\beta }\models \varphi (\gamma )).$$

This article is concerned with those ordinals α which are both ${\mathrm{\Sigma }}_1^1$-reflecting and locally countable, in the sense that every $\beta <\alpha$ is countable in $L_{\alpha }$; its purpose is to characterize these ordinals in game-theoretic terms.

Let us recall some background. An ordinal α is said to be admissible if $L_{\alpha }$ is a model of $\mathsf{KP}$, Kripke-Platek set theory. Every ${\mathrm{\Sigma }}_1^1$-reflecting ordinal is admissible and a limit of admissible ordinals. Given an admissible α, a subset A of α is α-recursively enumerable (α-r.e.) if it is ${\mathrm{\Sigma }}_1$-definable over $L_{\alpha }$; it is α-recursive if both A and $\alpha \setminus A$ are α-r.e. By usual methods, finite or ω-length sequences of elements of α which belong to $L_{\alpha }$ can be coded by single elements of α, so we may speak of α-recursive sets of sequences in α, etc.

A classical theorem due to Kripke and Platek states that the supremum of lengths of recursive wellorderings of ω is ${\omega }_1^{ck}$, the least admissible ordinal greater than ω. An admissible ordinal α is said to be Gandy if the supremum of α-recursive wellorderings of subsets of α is ${\alpha }^{+}$, the least admissible ordinal greater than α. Gostanian [9] has characterized the locally countable ${\mathrm{\Sigma }}_1^1$-reflecting ordinal as those which are not Gandy. By a theorem of Abramson and Sacks [2], ${({\omega }_{\omega }^L)}^{+}$ is Gandy, so not every Gandy ordinal is locally countable. Harrington (unpublished) has shown that, in L, the least uncountable Gandy ordinal has cardinality ${\omega }_1$.

The least ${\mathrm{\Sigma }}_1^1$-reflecting ordinal is commonly denoted by ${\sigma }_1^1$. Aczel and Richter [4] showed that this is the closure ordinal for ${\mathrm{\Sigma }}_1^1$ inductive definitions; by a theorem of Grilliot,¹ it is thus the closure ordinal for monotone ${\mathrm{\Sigma }}_1^1$ inductive definitions. Solovay (see Kechris [11] or Moschovakis [14]) showed that for every ${\mathrm{\Sigma }}_2^0$ game on $\mathbb{N}$, if Player I has a winning strategy for it, then she has one in $L_{{\sigma }_1^1}$ (and this is the least such ordinal); this has applications in the reverse mathematics of determinacy (see e.g., Tanaka [18]). Ackerman, Freer, and Lubarsky [3] show that ${\sigma }_1^1$ is the least ordinal which is not parallel-feedback computable. Other characterizations can be found in [7]. By a theorem of Aanderaa [1], ${\sigma }_1^1$ is greater than ${\pi }_1^1$, the least ${\mathrm{\Pi }}_1^1$-reflecting ordinal. ${\sigma }_1^1$ is stable precisely to the supremum of the ${\sigma }_1^1$-recursive wellorderings (see [5]).

We define the classes

,

, and

. The definition makes sense for arbitrary acceptable structures, though we restrict our attention to admissible initial segments of the constructible hierarchy. Let α be an admissible ordinal and fix a simple coding scheme assigning ordinals $〈{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n〉$ smaller than α to finite tuples $({\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n)$ of ordinals smaller than α. We say that a subset of α is

(“open-game definable”) if it can be defined over $L_{\alpha }$ by a formula of the form

$$\exists {\alpha }_0\in \alpha \forall {\alpha }_1\in \alpha \exists {\alpha }_2\in \alpha \dots \exists n\in \mathbb{N}\phi (〈{\alpha }_0,\dots ,{\alpha }_n〉,\beta ,x)$$

where $\beta <\alpha$ is a parameter and φ is ${\mathrm{\Sigma }}_1$. Here, the infinite quantifier string is interpreted in the natural way: (1) is said to hold if Player I has a winning strategy in the infinite game in which Players I and II alternate turns playing ordinals ${\alpha }_i<\alpha$, after which Player I wins if and only if $\phi (〈{\alpha }_0,\dots ,{\alpha }_n〉,\beta ,x)$ holds in $L_{\alpha }$, for some initial subsequence ${\alpha }_0,\dots ,{\alpha }_n$ of the play. We refer to these games as α-open games.

The

sets are those definable over $L_{\alpha }$ by a formula of the form

$$\exists {\alpha }_0\in \alpha \forall {\alpha }_1\in \alpha \exists {\alpha }_2\in \alpha \dots \forall n\in \mathbb{N}\phi (〈{\alpha }_0,\dots ,{\alpha }_n〉,\beta ,x)$$

where $\beta <\alpha$ is a parameter and φ is ${\mathrm{\Pi }}_1$. Observe that we assume (by adding dummy moves if necessary) that all α-open and all α-closed games begin with a move by Player I. These games are determined by the Gale-Stewart theorem [8], so the

sets are precisely those whose complements are

.

Finally, we may define the class of

(“clopen-game definable”) subsets of α to be those definable over $L_{\alpha }$ by a formula of the form (1) with the added condition that there exist ${\beta }^{\prime }<\alpha$ and a ${\mathrm{\Sigma }}_1$ formula ${\phi }^{\prime }$ such that for every $({\alpha }_0,{\alpha }_1,\dots )\in {\alpha }^{\omega }$ and every $x\in L_{\alpha }$, exactly one of the following holds:

• there exist $n\in \mathbb{N}$ such that $\phi (〈{\alpha }_0,\dots ,{\alpha }_n〉,\beta ,x)$, or

• there exist $n\in \mathbb{N}$ such that ${\phi }^{\prime }(〈{\alpha }_0,\dots ,{\alpha }_n〉,{\beta }^{\prime },x)$.

Associated to each of these sets is an α-clopen game in which Players I and II alternate turns playing ordinals ${\alpha }_i<\alpha$, after which Player I wins if and only if $\phi (〈{\alpha }_0,\dots ,{\alpha }_n〉,\beta ,x)$ holds in $L_{\alpha }$ for some initial subsequence ${\alpha }_0,\dots ,{\alpha }_n$ of the play. The formulae φ and ${\phi }^{\prime }$, as well as the parameters $\beta ,{\beta }^{\prime }$, and x, can be regarded as the “rules” of the game, and as x varies, the rules might differ (although they are given uniformly). By the clopenness condition, Player II wins if and only if ${\phi }^{\prime }(〈{\alpha }_0,\dots ,{\alpha }_n〉,{\beta }^{\prime },x)$ holds in $L_{\alpha }$ for some initial subsequence ${\alpha }_0,\dots ,{\alpha }_n$ of the play. We refer to

$$\{({\alpha }_0,{\alpha }_1,\dots )\in {\alpha }^{\omega }:\exists n\in \mathbb{N}{L}_{\alpha }\models \phi (〈{\alpha }_0,\dots ,{\alpha }_n〉,\beta ,x)\}$$

as the payoff of the game. The main relevant property of α-clopen games is that every time one is played, the outcome is decided after only a finite number of turns.

Remark 1

If $\alpha =\omega$, then α-open (closed, clopen) games are equivalent to the usual open (closed, clopen) games on natural numbers, and their payoff sets are open (closed, clopen) in the product space ${\omega }^{\omega }$.

We shall prove the following theorem:

Theorem 2

Let α be a locally countable admissible ordinal. The following are equivalent:

• α is ${\mathit{\Sigma }}_1^1$-reflecting;

• over $L_{\alpha }$,

  .