The inclusion relations of the countable models of set theory are all isomorphic

TL;DR

This paper proves that all countable models of set theory, whether well-founded or not, have isomorphic inclusion structures, which are characterized as countable saturated models of set-theoretic mereology.

Contribution

It establishes that the inclusion relations of countable models of set theory are all isomorphic and characterizes these structures as saturated models of set-theoretic mereology.

Findings

All countable models' inclusion structures are isomorphic.

These structures are exactly the countable saturated models of set-theoretic mereology.

Weak set theories, including finite set theory, suffice for the results.

Abstract

The structures $\langle M,\subseteq^M\rangle$ arising as the inclusion relation of a countable model of sufficient set theory $\langle M,\in^M\rangle$, whether well-founded or not, are all isomorphic. These structures $\langle M,\subseteq^M\rangle$ are exactly the countable saturated models of the theory of set-theoretic mereology: an unbounded atomic relatively complemented distributive lattice. A very weak set theory suffices, even finite set theory, provided that one excludes the $\omega$-standard models with no infinite sets and the $\omega$-standard models of set theory with an amorphous set. Analogous results hold also for class theories such as G\"odel-Bernays set theory and Kelley-Morse set theory.