The subset relation and 2-stratified sentences in set theory and class theory

Abstract

Hamkins and Kikuchi (2016, 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. This paper identifies the minimum subsystem of $\mathrm{ZF}$, $\mathrm{BAS}$, that ensures that the definable subset ordering of the universe interprets a complete theory, and classifies the structures that can be realised as the subset relation in a model of this set theory. Extending and refining Hamkins and Kikuchi's result for class theory, a complete extension, ${\mathrm{IABA}}_{\mathrm{Ideal}}$, of the theory of infinite atomic boolean algebras and a minimum subsystem, ${\mathrm{BAC}}^{+}$, of $\mathrm{NBG}$ are identified with the property that if $M$ is a model of ${\mathrm{BAC}}^{+}$, then $⟨M,S^M,{\subseteq }^M⟩$ is a model of ${\mathrm{IABA}}_{\mathrm{Ideal}}$, where M is the underlying set of $M$, $S^M$ is the unary predicate that distinguishes sets from classes and ${\subseteq }^M$ is the definable subset relation. These results are used to show that that $\mathrm{BAS}$ decides every 2-stratified sentence of set theory and ${\mathrm{BAC}}^{+}$ decides every 2-stratified sentence of class theory.