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

Abstract

The structures $⟨M,{\subseteq }^M⟩$ arising as the inclusion relation of a countable model of sufficient set theory $⟨M,{\in }^M⟩$, whether well-founded or not, are all isomorphic. These structures $⟨M,{\subseteq }^M⟩$ 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 Gödel-Bernays set theory and Kelley-Morse set theory.

arXiv comment20 pages. Commentary can be made on the first author's blog at http://jdh.hamkins.org/inclusion-relations-are-all-isomorphic