Borel Complexity and Potential Canonical Scott Sentences

TL;DR

This paper introduces HC-forcing invariant formulas and a cardinality measure for $L_{,}$ sentences, analyzing the complexity of models of certain theories and establishing bounds on their classification difficulty.

Contribution

It defines a new notion of cardinality for $L_{,}$ sentences and uses it to analyze and compare the complexity of classes of models, including Borel and non-Borel cases.

Findings

The cardinal ||Phi|| bounds the complexity of (Mod(Phi), iso).

Certain theories have non-Borel but not Borel complete model classes.

Borel complexity is bounded by I_{,}(Phi) < 6_{_1}.

Abstract

We define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Phi|| for sentences Phi of $L_{\omega_1,\omega}$, which counts the number of sentences of $L_{\infty,\omega}$ that, in some forcing extension, become a canonical Scott sentence of a model of Phi. We show this cardinal bounds the complexity of (Mod(Phi), iso), the class of models of Phi with universe omega, by proving that (Mod(Phi),iso) is not Borel reducible to (Mod(Psi),iso) whenever ||Psi|| < ||Phi||. Using these tools, we analyze the complexity of the class of countable models of four complete, first-order theories T for which (Mod(T),iso) is properly analytic, yet admit very different behavior. We prove that both `Binary splitting, refining equivalence relations' and Koerwien's example of an eni-depth 2, omega-stable theory have (Mod(T),iso) non-Borel, yet neither is Borel complete. We give a slight modification of Koerwien's example that also is omega-stable, eni-depth 2, but is Borel complete. Additionally, we prove that I_{\infty,\omega}(Phi)<\beth_{\omega_1} whenever (Mod(Phi),iso) is Borel.