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Let $(P_r)_{r \in \mathbb{R}}$ be an $\mathbb{R}$-indexed family of propositional variables. Let $\mathcal{L}$ be the collection of all propositional sentences formed from the variables $(P_r)_{r \in \mathbb{R}}$. We can clearly regard $\mathcal{L}$ as a Polish space in a natural way. A theory $T$ is a subset of $\mathcal{L}$. A truth assignment is a function $f: \mathbb{R} \to \{\top,\bot\}$. We say that a truth assignment $f$ satisfies a theory $T$ if it does so in the standard sense of propositional logic. We'll say that a theory $T$ is Borel if it is Borel as a subset of $\mathcal{L}$.

Broadly speaking, I'm interested in characterizing when a Borel or otherwise topologically tame theory has a topologically tame completion. Three versions of this question immediately come to mind.

Questions. When is a consistent Borel theory satisfied by a Borel measurable truth assignment? A Baire measurable truth assignment? A Lebesgue measurable truth assignment?

We could of course ask similar questions about $\Sigma^1_1$ or $\Pi^1_1$ theories. In the case of $\Sigma^1_1$ theories at least, I believe we can reduce this to the Borel case, although please tell me if I've missed something in this argument.

Proposition. For any consistent $\Sigma^1_1$ theory $T$, there is a consistent Borel theory $T' \supseteq T$.

Proof. Let $C \subseteq 2^{\mathcal{L}}$ be the collection of consistent theories. It is fairly direct to show that $C$ is $\Pi^1_1$ on $\Sigma^1_1$ (i.e., for any Polish space $X$ and any $\Sigma^1_1$ set $A \subseteq X \times \mathcal{L}$, the set $\{x \in X : A_x \in C\}$ is $\Pi^1_1$). Therefore, by the first reflection theorem there is a Borel $T' \in C$ with $T' \supseteq T$. $\square$

One thing to note about this proposition is that it means we can always extend a consistent Borel theory to a consistent Borel theory that is closed under logical consequence. For any Borel theory $T$, the set of sentences logically entailed by $T$ is $\Sigma^1_1$. This allows us to build a chain $(T_i)_{i<\omega}$ of consistent Borel theories with $T_0 = T$ such that the logical consequences of $T_{i+1}$ are contained in $T_i$. The union is then a consistent Borel theory closed under logical consequence.

I think it might be possible to make a partial statement using Lecomte's infinite arity generalization of the Kechris-Solecki-Todorcevic dichotomy. Specifically, given a Borel theory $T$, we can define a $\Sigma^1_1$ $\omega$-ary hypergraph $A \subseteq (\mathbb{R}\times \{\top,\bot\})^\omega$ where $\alpha \in (\mathbb{R} \times \{\top,\bot\})^\omega$ is in $A$ if and only if $\alpha$ is non-constant and codes a partial truth assignment that is inconsistent with $T$. This is clearly a $\Sigma^1_1$ relation. Lecomte's result then implies that if a certain Borel hypergraph does not have a Baire measurable homomorphism into our hyergraph $A$, then $A$ has a countable Borel coloring. Each monochromatic set in this coloring is an anticlique in the graph, and therefore corresponds to a partial truth assignment that is consistent with $T$. Applying the Baire category theorem or basic facts about the Lebesgue measure then implies that we can find truth assignments that are either Baire measurable in some open set or Lebesgue measurable on some set of positive measure. One could then try to iterate this argument, using the partial truth assignment to make a bigger consistent Borel theory.

At the moment I'm unsatisfied with this approach though because, whereas Lecomte's result is sharp in its own context, we're only using part of it here. Really, we have a finitary hypergraph with arbitrarily high arity edges (since consistency is a finitary property), and really we're just after an anticlique, not a coloring. I haven't been able to find anything that seemed directly relevant to me in the descriptive set theoretic literature.

Similar questions have been investigated in a first-order context by Hjorth and Nies, but I'm not sure how directly applicable their work is to my questions here.

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    $\begingroup$ I've added a link to some slides with an exposition of the first reflection theorem you mention, for those (like me!) not familiar with it already. (There are too many "reflection theorems" in logic to be guaranteed easy googling IMO.) Also, almost-trivial aside: Lusin's separation theorem implies that every consistent $\Sigma^1_1$ set of atomic-or-negated-atomic sentences admits a Borel truth assignment. $\endgroup$ Jun 19, 2022 at 3:19
  • $\begingroup$ @NoahSchweber Thanks. Incidentally, the reference I was using for the first reflection theorem is Theorem 35.10 in Classical Descriptive Set Theory by Kechris. As is implicit in the name, there is a second reflection theorem, but I haven't really wrapped my head around how one might use it in this context. $\endgroup$ Jun 19, 2022 at 6:31

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