[bibtex key=”HamkinsKikuchi:The-inclusion-relations-of-the-countable-models-of-set-theory-are-all-isomorphic”]

Abstract. The structures $\langle M,\newcommand\of{\subseteq}\of^M\rangle$ 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 $\langle M,\of^M\rangle$ 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 Gödel-Bernays set theory and Kelley-Morse set theory.

Set-theoretic mereology is the study of the inclusion relation $\of$ as it arises within set theory. In any set-theoretic context, with the set membership relation $\in$, one may define the corresponding inclusion relation $\of$ and investigate its properties. Thus, every model of set theory $\langle M,\in^M\rangle$ gives rise to a corresponding model of set-theoretic mereology $\langle M,\of^M\rangle$, the reduct to the inclusion relation.

In our previous article,

J. D. Hamkins and M. Kikuchi, Set-theoretic mereology, Logic and Logical Philosophy, special issue “Mereology and beyond, part II”, vol. 25, iss. 3, pp. 1-24, 2016.

we had identified exactly the complete theory of these mereological structures $\langle M,\of^M\rangle$. Namely, if $\langle M,\in^M\rangle$ is a model of set theory, even for extremely weak theories, including set theory without the infinity axiom, then the corresponding mereological reduct $\langle M,\of^M\rangle$ is an unbounded atomic relatively complemented distributive lattice. We call this the theory of set-theoretic mereology. By a quantifier-elimination argument that we give in our earlier paper, partaking of Tarski’s Boolean-algebra invariants and Ersov’s work on lattices, this theory is complete, finitely axiomatizable and decidable.  We had proved among other things that $\in$ is never definable from $\of$ in any model of set theory and furthermore, some models of set-theoretic mereology can arise as the inclusion relation of diverse models of set theory, with different theories. Furthermore, we proved that $\langle\text{HF},\subseteq\rangle\prec\langle V,\subseteq\rangle$.

After that work, we found it natural to inquire:

Question. Which models of set-theoretic mereology arise as the inclusion relation $\of$ of a model of set theory?

More precisely, given a model $\langle M,\newcommand\sqof{\sqsubseteq}\sqof\rangle$ of set-theoretic mereology, under what circumstances can we place a binary relation $\in^M$ on $M$ in such a way that $\langle M,\in^M\rangle$ is a model of set theory and the inclusion relation $\of$ defined in $\langle M,\in^M\rangle$ is precisely the given relation $\sqof$? One can view this question as seeking a kind of Stone-style representation of the mereological structure $\langle M,\sqof\rangle$, because such a model $M$ would provide a representation of $\langle M,\sqof\rangle$ as a relative field of sets via the model of set theory $\langle M,\in^M\rangle$.

A second natural question was to wonder how much of the theory of the original model of set theory can be recovered from the mereological reduct.

Question. If $\langle M,\of^M\rangle$ is the model of set-theoretic mereology arising as the inclusion relation $\of$ of a model of set theory $\langle M,\in^M\rangle$, what part of the theory of $\langle M,\in^M\rangle$ is determined by the structure $\langle M,\of^M\rangle$?

In the case of the countable models of ZFC, these questions are completely answered by our main theorems.

Main Theorems.

1. All countable models of set theory $\langle M,\in^M\rangle\models\text{ZFC}$ have isomorphic reducts $\langle M,\of^M\rangle$ to the inclusion relation.
2. The same holds for models of considerably weaker theories such as KP and even finite set theory, provided one excludes the $\omega$-standard models without infinite sets and the $\omega$-standard models having an amorphous set.
3. These inclusion reducts $\langle M,\of^M\rangle$ are precisely the countable saturated models of set-theoretic mereology.
4. Similar results hold for class theory: all countable models of Gödel-Bernays set theory have isomorphic reducts to the inclusion relation, and this reduct is precisely the countably infinite saturated atomic Boolean algebra.

Specifically, we show that the mereological reducts $\langle M,\of^M\rangle$ of the models of sufficient set theory are always $\omega$-saturated, and from this it follows on general model-theoretic grounds that they are all isomorphic, establishing statements (1) and (2). So a countable model $\langle M,\sqof\rangle$ of set-theoretic mereology arises as the inclusion relation of a model of sufficient set theory if and only if it is $\omega$-saturated, establishing (3) and answering the first question. Consequently, in addition, the mereological reducts $\langle M,\of^M\rangle$ of the countable models of sufficient set theory know essentially nothing of the theory of the structure $\langle M,\in^M\rangle$ from which they arose, since $\langle M,\of^M\rangle$ arises equally as the inclusion relation of other models $\langle M,\in^*\rangle$ with any desired sufficient alternative set theory, a fact which answers the second question. Our analysis works with very weak set theories, even finite set theory, provided one excludes the $\omega$-standard models with no infinite sets and the $\omega$-standard models with an amorphous set, since the inclusion reducts of these models are not $\omega$-saturated. We also prove that most of these results do not generalize to uncountable models, nor even to the $\omega_1$-like models.

Our results have some affinity with the classical results in models of arithmetic concerned with the additive reducts of models of PA. Restricting a model of set theory to the inclusion relation $\of$ is, after all, something like restricting a model of arithmetic to its additive part. Lipshitz and Nadel (1978) proved that a countable model of Presburger arithmetic (with $+$ only) can be expanded to a model of PA if and only if it is computably saturated. We had hoped at first to prove a corresponding result for the mereological reducts of the models of set theory. In arithmetic, the additive reducts are not all isomorphic, since the standard system of the PA model is fully captured by the additive reduct. Our main result for the countable models of set theory, however, turned out to be stronger than we had expected, since the inclusion reducts are not merely computably saturated, but fully $\omega$-saturated, and this is why they are all isomorphic. Meanwhile, Lipshitz and Nadel point out that their result does not generalize to uncountable models of arithmetic, and similarly ours also does not generalize to uncountable models of set theory.

The work leaves the following question open:

Question. Are the mereological reducts $\langle M,\of^M\rangle$ of all the countable models $\langle M,\in^M\rangle$ of ZF with an amorphous set all isomorphic?

We expect the answer to come from a deeper understanding of the Tarski-Ersov invariants for the mereological structures combined with knowledge of models of ZF with amorphous sets.

This is joint work with Makoto Kikuchi.