In my recent paper with Makoto Kikuchi,

• J. D. Hamkins and M. Kikuchi, The inclusion relations of the countable models of set theory are all isomorphic. manuscript under review. (arχiv)

we proved the following theorem.

Theorem. All countable models $\langle M,\in^M\rangle\models\text{ZFC}$, whether well-founded or not, have the same inclusion relation $\langle M,\subseteq^M\rangle$, up to isomorphism.

And the same is true for much weaker theories, such as KP and even finite set theory, provided that one excludes the $\omega$-standard models without any infinite sets and also the $\omega$-standard models with amorphous sets.

The proof proceeds by proving that for most of the models of set theory $\langle M,\in^M\rangle$, the corresponding inclusion relation $\langle M,\subseteq\rangle$ is an $\omega$-saturated model of what we have called set-theoretic mereology, which is the theory of an unbounded atomic relatively complemented distributed lattice. Those are the basic facts of $\subseteq$ in set theory and that is a complete, finitely axiomatizable, decidable theory. Since the theory is complete, and the resulting models are $\omega$-saturated, it follows by the back-and-forth method that all countable saturated models are isomorphic.

But the possibility of amorphous sets throws a wrench in the works. For most models of set theory, we prove that the inclusion relation is $\omega$-saturated; but when the model has an amorphous set, then it is not $\omega$-saturated, and so this is an irrirating obstacle to the general phenomenon. What we want to know is whether the ZF models provide just one or many different isomorphism types for the inclusion relation.

Question. Do all countable models of ZF with an amorphous set have isomorphic inclusion relations?

Kikuchi and I state in the paper that we believe that the answer to this question will come from an understanding of the Tarski/Ersov invariants combined with a knowledge of the models of ZF with amorphous sets.

In my recent paper with Makoto Kikuchi,

• J. D. Hamkins and M. Kikuchi, The inclusion relations of the countable models of set theory are all isomorphic. manuscript under review. (arχiv)

we proved the following theorem.

Theorem. All countable models $\langle M,\in^M\rangle\models\text{ZFC}$, whether well-founded or not, have the same inclusion relation $\langle M,\subseteq^M\rangle$, up to isomorphism.

And the same is true for much weaker theories, such as KP and even finite set theory, provided that one excludes the $\omega$-standard models without any infinite sets and also the $\omega$-standard models with amorphous sets.

The proof proceeds by proving that for most of the models of set theory $\langle M,\in^M\rangle$, the corresponding inclusion relation $\langle M,\subseteq\rangle$ is an $\omega$-saturated model of what we have called set-theoretic mereology, which is the theory of an unbounded atomic relatively complemented distributed lattice. Those are the basic facts of $\subseteq$ in set theory and that is a complete, finitely axiomatizable, decidable theory. Since the theory is complete, and the resulting models are $\omega$-saturated, it follows by the back-and-forth method that all countable saturated models are isomorphic.

But the possibility of amorphous sets throws a wrench in the works. For most models of set theory, we prove that the inclusion relation is $\omega$-saturated; but when the model is $\omega$-standard and has an amorphous set, then it is not $\omega$-saturated, and so this is an irrirating obstacle to the general phenomenon. What we want to know is whether the ZF models provide just one or many different isomorphism types for the inclusion relation.

Question. Do all countable $\omega$-standard models of ZF with an amorphous set have isomorphic inclusion relations?

Kikuchi and I state in the paper that we believe that the answer to this question will come from an understanding of the Tarski/Ersov invariants combined with a knowledge of the models of ZF with amorphous sets.