Borel complexity and potential canonical Scott sentences
Volume 239 / 2017
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 $\|\varPhi\|$ for sentences $\varPhi$ 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 $\varPhi$. We show this cardinal bounds the complexity of $(\operatorname{Mod}(\varPhi), {\cong})$, the class of models of $\varPhi$ with universe $\omega$, by proving that $(\operatorname{Mod}(\varPhi),{\cong})$ is not Borel reducible to $(\operatorname{Mod}(\varPsi),{\cong})$ whenever $\|\varPsi\| \lt \|\varPhi\|$. Using these tools, we analyze the complexity of the class of countable models of four complete, first order theories $T$ for which $(\operatorname{Mod}(T),{\cong})$ is properly analytic, yet admit very different behavior. We prove that both “binary splitting, refining equivalence relations” and Koerwien’s example (2011) of an eni-depth 2, $\omega$-stable theory have $(\operatorname{Mod}(T),{\cong})$ non-Borel, yet neither is Borel complete. We give a slight modification of Koerwien’s example that is also $\omega$-stable, eni-depth 2, but is Borel complete. Additionally, we prove that $I_{\infty\omega}(\varPhi) \lt \beth_{\omega_1}$ whenever $(\operatorname{Mod}(\varPhi),{\cong})$ is Borel.
