Borel Complexity and Potential Canonical Scott Sentences

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 ||Φ|| for sentences Φ of Lω1,ω, which counts the number of sentences of L∞,ω that, in some forcing extension, become a canonical Scott sentence of a model of Φ. We show this cardinal bounds the complexity of (Mod(Φ),∼=), the class of models of Φ with universe ω, by proving that (Mod(Φ),∼=) is not Borel reducible to (Mod(Ψ),∼=) whenever ||Ψ|| < ||Φ||. Using these tools, we analyze the complexity of the class of countable models of four complete, first-order theories T for which (Mod(T ),∼=) is properly analytic, yet admit very different behavior. We prove that both ‘Binary splitting, refining equivalence relations’ and Koerwien’s example [14] of an eni-depth 2, ω-stable theory have (Mod(T ),∼=) non-Borel, yet neither is Borel complete. We give a slight modification of Koerwien’s example that also is ω-stable, eni-depth 2, but is Borel complete. Additionally, we prove that I∞,ω(Φ) < iω1 whenever (Mod(Φ),∼=) is Borel.