1. The inclusion relations of the countable models of set theory are all isomorphic
- Author
-
Hamkins, Joel David and Kikuchi, Makoto
- Subjects
Mathematics - Logic - 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., Comment: 20 pages. Commentary can be made on the first author's blog at http://jdh.hamkins.org/inclusion-relations-are-all-isomorphic
- Published
- 2017