Journal of Symbolic Logic

Countable Structures, Ehrenfeucht Strategies, and Wadge Reductions

Tom Linton

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For countable structures $\mathfrak{U}$ and $\mathfrak{B}$, let $\mathfrak{U}\overset{\alpha}{\rightarrow}\mathfrak{B}$ abbreviate the statement that every $\Sigma^0_\alpha (\mathbf{L}_{\omega_1,\omega})$ sentence true in $\mathfrak{U}$ also holds in $\mathfrak{B}$. One can define a back and forth game between the structures $\mathfrak{U}$ and $\mathfrak{B}$ that determines whether $\mathfrak{U}\overset{\alpha}{\rightarrow}\mathfrak{B}$. We verify that if $\theta$ is an $\mathbf{L}_{\omega,\omega}$ sentence that is not equivalent to any $\mathbf{L}_{\omega,\omega} \Sigma^0_n$ sentence, then there are countably infinite models $\mathfrak{U}$ and $\mathfrak{B}$ such that $\mathfrak{U} \vDash \theta, \mathfrak{B} \vDash \neg \theta$, and $\mathfrak{U}\overset{n}{\rightarrow}\mathfrak{B}$. For countable languages $\mathscr{L}$ there is a natural way to view $\mathscr{L}$ structures with universe $\omega$ as a topological space, $X_\mathscr{L}$. Let $\lbrack\mathfrak{U}\rbrack = \{\mathfrak{B} \in X_\mathscr{L}\mid\mathfrak{B} \cong \mathfrak{U}\}$ denote the isomorphism class of $\mathfrak{U}$. Let $\mathfrak{U}$ and $\mathfrak{B}$ be countably infinite nonisomorphic $\mathscr{L}$ structures, and let $C \subseteq \omega^\omega$ be any $\Pi^0_\alpha$ subset. Our main result states that if $\mathfrak{U}\overset{\alpha}{\rightarrow}\mathfrak{B}$, then there is a continuous function $f: \omega^\omega \rightarrow X_\mathscr{L}$ with the property that $x \in C \Rightarrow f(x) \in \lbrack\mathfrak{U}\rbrack$ and $x \notin C \Rightarrow f(x) \in \lbrack\mathfrak{B}\rbrack$. In fact, for $\alpha \leq 3$, the continuous function $f$ can be defined from the $\overset{\alpha}{\rightarrow}$ relation.

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J. Symbolic Logic, Volume 56, Issue 4 (1991), 1325-1348.

First available in Project Euclid: 6 July 2007

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Linton, Tom. Countable Structures, Ehrenfeucht Strategies, and Wadge Reductions. J. Symbolic Logic 56 (1991), no. 4, 1325--1348.

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