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

TL;DR

This paper explores the logical properties of subset relations in set and class theories, identifying minimal subsystems that decide all 2-stratified sentences and classifying the structures they can realize.

Contribution

It identifies minimal subsystems of ZF and NBG ensuring the definable subset relations form specific algebraic structures and proves these theories decide all 2-stratified sentences.

Findings

BAS ensures the subset relation forms an atomic unbounded relatively complemented distributive lattice.

BAC ensures the subset relation forms an infinite atomic Boolean algebra.

Theories like IABA_Ideal and BAC+ decide all 2-sentences in their respective languages.

Abstract

Hamkins and Kikuchi (2016 and 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. If $\mathcal{M}$ is a model of set theory, then $\langle M, \subseteq^\mathcal{M} \rangle$ is an atomic unbounded relatively complemented distributive lattice. If $\mathcal{M}$ is model of class theory, then $\langle M, \subseteq^\mathcal{M} \rangle$ is an infinite atomic boolean algebra. We identify the minimal subsystem of $\mathrm{ZF}$, $\mathrm{BAS}$, that ensures that the definable subset relation is an atomic unbounded relatively complemented distributive lattice and classify the atomic unbounded relatively complemented distributive lattices that can be realised as a subset relations of this theory. The fact that the theory of atomic unbounded relatively complemented distributive lattices is complete is used to show that $\mathrm{BAS}$ decides every $2$-stratified sentence of set theory. We also identify the minimal subsystem of $\mathrm{NBG}$, $\mathrm{BAC}$, that ensures that the definable subset relation is an infinite atomic Boolean algebra. We show that there is a complete extension, $\mathrm{IABA}_\mathrm{Ideal}$, of the theory of infinite atomic boolean algebras and an extension $\mathrm{BAC}^+$ of $\mathrm{BAC}$ corresponding to the minimal theory such that if $\mathcal{M}$ is a model of $\mathrm{BAC}^+$, then $\langle M, \mathcal{S}^\mathcal{M}, \subseteq^\mathcal{M} \rangle$ is a model of $\mathrm{IABA}_{\mathrm{Ideal}}$, where $\mathcal{S}^\mathcal{M}$ is a unary predicate that distinguishes sets from classes. This is used to show that $\mathrm{BAC}^+$, a subsystem of $\mathrm{NBG}$, decides every $2$-sentence in the language of class theory that includes a unary predicate distinguishing sets from classes.