Borel complexity and potential canonical Scott sentences

Douglas Ulrich ¹   ; Richard Rast ¹   ; Michael C. Laskowski ¹

¹ Department of Mathematics University of Maryland College Park, MD 20742, U.S.A.

Fundamenta Mathematicae, Tome 239 (2017) no. 2, pp. 101-147

Résumé

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.

Détail

DOI : 10.4064/fm326-11-2016

Keywords: define investigate hc forcing invariant formulas set theory whose interpretations hereditarily countable sets well behaved under forcing extensions leads naturally notion cardinality varphi sentences varphi omega omega which counts number sentences infty omega forcing extension become canonical scott sentence model varphi cardinal bounds complexity operatorname mod varphi cong class models varphi universe omega proving operatorname mod varphi cong borel reducible operatorname mod varpsi cong whenever varpsi varphi using these tools analyze complexity class countable models complete first order theories which operatorname mod cong properly analytic yet admit different behavior prove binary splitting refining equivalence relations koerwien example eni depth omega stable theory have operatorname mod cong non borel yet neither borel complete slight modification koerwien example omega stable eni depth borel complete additionally prove infty omega varphi beth omega whenever operatorname mod varphi cong borel