# Complexity and classification of countable models

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## Finitely generated models

This question is about the complexity of the classification problem for finitely generated models of a given completion $T$ of $PA$.

Let us denote the space of finitely generated models of $T$ by $X^{fg}_T$. (More precisely: if $A$ is a fixed countable set, then we can identify the set of models of $T$ whose underlying set is $A$ with a subset of $A^{A^6}$, where each model is represented by its addition and multiplication functions.) Letting $\cong^{fg}_T$ denote the isomorphism relation on $X^{fg}_T$, it is not hard to see that $\cong^{fg}_T$ is Borel, considered as a subset of $X^{fg}_T\times X^{fg}_T$.

The complexity of $\cong^{fg}_T$ can be analyzed more closely using the concept of Borel reducibility. Here, if $E,F$ are equivalence relations on standard Borel spaces $X,Y$, then $E$ is said to be Borel reducible to $F$ iff there is a Borel function $f\colon X\to Y$ such that $x\mathrel{E}x'\iff f(x)\mathrel{F}f(x')$ See  for an introduction and survey of this subject of study.

It was shown in  that $\cong^{fg}_T$ is Borel reducible to $E_\infty$, the universal countable equivalence relation. It was furthermore shown that $E_0$, the almost equality relation on the set of infinite binary sequences, is Borel reducible to $\cong^{fg}_T$.

Problem: What is the precise complexity of $\cong^{fg}_T$? There is a wide gap between $E_0$ and $E_\infty$. Is $\cong^{fg}_T$ bireducible with $E_\infty$? Is $\cong^{fg}_T$ bireducible with $E_0$?

It should be remarked that the classification of many other sorts of finitely generated structures lies at the level of $E_\infty$. For instance, the classification of locally finite graphs, finitely branching trees, finitely generated groups, and fields of finite transcendence degree all have been shown to be bireducible with $E_\infty$. Some of these arguments are worked out in Chapter 13 of .