The number of models of a fixed Scott rank, for a counterexample to the analytic Vaught conjecture

TL;DR

The paper investigates the number of models with fixed Scott rank in counterexamples to the analytic Vaught conjecture, showing a club set where the number of models remains constant at each rank.

Contribution

It establishes a new structural property of counterexamples to the analytic Vaught conjecture regarding the distribution of models across Scott ranks.

Findings

Existence of a club set where the number of models of each Scott rank is constant.

Counterexamples with a fixed number of models at Scott rank ω₁ have a predictable distribution across a club set.

Provides insight into the structure of models in the context of the analytic Vaught conjecture.

Abstract

We show that if $\gamma \in \omega \cup \{\aleph_{0}\}$ and $\mathcal{A}$ is a counterexample to the analytic Vaught conjecture having exactly $\gamma$ many models of Scott rank $\omega_{1}$, then there exists a club $C \subseteq \omega_{1}$ such that $\mathcal{A}$ has exactly $\gamma$ many models of Scott rank $\alpha$, for each $\alpha \in C$.

Full text

Throughout this note $\tau$ represents a countable relational vocabulary. The set of $\tau$-structures with domain $\omega$ is naturally seen as a Polish space $X_{\tau}$, where a basic open set is given by the set of structures in which $R(i_{0},\ldots,i_{n-1})$ holds, for $R$ an $n$-ary relation symbol from $\tau$ and $i_{0},\ldots,i_{n-1}\in\omega$ (see Section 11.3 of [1], for instance). Given a sentence $\phi\in\mathcal{L}_{\aleph_{1},\aleph_{0}}(\tau)$, the set of models of $\phi$ with domain $\omega$ is a Borel subset of $X_{\tau}$. By a theorem of Lopez-Escobar [4], every Borel subset of $X_{\tau}$ which is closed under isomporphism is also the set of models (with domain $\omega$) of some $\mathcal{L}_{\aleph_{1},\aleph_{0}}(\tau)$ sentence.

We call the following (false) statement the analytic Vaught conjecture: for every countable relational vocabulary $\tau$, every analytic subset of $X_{\tau}$ which is closed under isomorphism and contains uncountably many nonisomorphic structures contains a perfect set of nonisomorphic structures. Steel [11] presents two counterexamples to this statement, one due to H. Friedman and the other to K. Kunen.

Given a $\tau$-structure $M$, we let $\mathrm{SP}_{\alpha}(M)$ denote the Scott process of $M$ of length $\alpha$, as defined in [3] (this is essentially the same as the standard definition appearing in [9, 5, 6]; we assume some familiarity with [3] in the arguments below, but expect that familiarity with the classical Scott analysis will suffice). Scott’s Isomorphism Theorem [9] (rephrased) says that if $\alpha$ is a (necessarily countable) ordinal and $M$ and $N$ are countable $\tau$-structures of Scott rank at most $\alpha$, then $M$ and $N$ are isomorphic if and only if $\mathrm{SP}_{\alpha+1}(M)=\mathrm{SP}_{\alpha+1}(N)$.

Given a set $\mathcal{A}\subseteq X_{\tau}$, we let $\mathcal{A}^{*}$ denote the class of (ground model, but possibly uncountable) $\tau$-structures $M$ which are isomorphic to an element of the reinterpretation of $\mathcal{A}$ in any (equivalently, every, by $\Sigma^{1}_{1}$-absoluteness) outer model in which $M$ is countable. If $\mathcal{A}$ is the set of $\tau$-structures on $\omega$ satisfying a sentence $\phi$ of $\mathcal{L}_{\aleph_{1},\aleph_{0}}(\tau)$, then $\mathcal{A}^{*}$ as defined above is simply the class of models of $\phi$.

For an ordinal $\alpha$, we let $\mathrm{SP}_{\alpha}(\mathcal{A})$ denote the set of the Scott processes of length $\alpha$ for structures in $\mathcal{A}^{*}$. If $\mathcal{A}$ is a counterexample to the analytic Vaught conjecture, then $|\mathrm{SP}_{\alpha}(\mathcal{A})|\leq|\alpha|$ (for $\alpha<\omega_{1}$ this follows by an induction argument using the Perfect Set Property for analytic sets; considering of a forcing extension via ${\rm Col}(\omega,\alpha)$ completes the argument for $\alpha\geq\omega_{1}$).

We also let $\mathcal{A}_{\alpha}$ denote respectively the class of structures $\mathcal{A}^{*}$ of Scott rank $\alpha$. The following well-known fact (slightly restated here) appears as Corollary 10.2 in [3].

The proof of Fact 0.1 given in [3] shows the following. Similar arguments appear in Section 1 of [2] and Chapter 32 of [7].

Theorem 0.2 below can also be proved using material from [10].

Fix $Y\in\mathcal{Y}$, let $\alpha=Y\cap\omega_{1}$ and let $P$ be the transitive collapse of $Y$. By Theorem 0.1, $\mathrm{SP}_{\alpha+1}(\mathcal{A})=\mathrm{SP}_{\alpha+1}(\mathcal{A})^{P}$. Suppose toward a contradiction that there exists an $N\in\mathcal{A}_{\alpha}\setminus\mathcal{M}_{Y}$. Then $\mathrm{SP}_{\alpha+1}(N)\in P$. Proposition 5.19 of [3] then implies that the $\delta$-th level of $\mathrm{SP}_{\delta+1}(N)$ amalgamates, as defined in Definition 5.16 of [3]. Since amalgamation is a first order property it is witnessed in $P$. It follows from Propositon 7.10 of [3] that there is a model of $\mathrm{SP}_{\delta+1}(N)$ in $P$, contradicting the elementarity of the collapse and the assumed property of $\mathcal{M}$. ∎

The proofs of Theorems 0.1 and 0.2 can be used to prove the following variation, which we leave to the interested reader : there is fragment $T$ of ZFC such that, if $x\subseteq\omega$ is a code for an analytic class $\mathcal{A}$ of $\tau$-structures then for any transitive model $P$ of $T$ containing $x$ and any ordinal $\alpha$ in $P$, if $\mathrm{SP}_{\alpha}(\mathcal{A})$ is countable then $\mathrm{SP}_{\alpha}(\mathcal{A})=\mathrm{SP}_{\alpha}(\mathcal{A})^{P}$, and, if in addition $\alpha<\omega_{2}^{P}$, then every structure in $\mathcal{A}^{*}$ of Scott rank $\alpha$ is isomorphic to one in $P$.