Resource request on "$\in$-homomorphisms" in Set Theory

Question

Very loosely put, this is the intuitive idea behind an $\in$-homomorphism:

Let $\mathcal{U}$ and $\mathcal{W}$ be universes of sets. A function $f \colon \mathcal{U} \to \mathcal{W}$ is said to be an $\in$-homomorphism if $f(X) = \{f(T) \in \mathcal{W} | T \in X \}$ for all $X \in \mathcal{U} $

Thomas Jech briefly mentions this concept in his book Set Theory on page 250 and 251. We note that he makes use of ZFA, an alternative axiomatisation of Set Theory which allows non-set objects called atoms.

Could anyone perhaps point me towards any other resource that also mentions, or perhaps goes deeper into this concept?

Ideally, I would like a resource, text or paper that also makes use of the ZFA axiomatisation, but I would also appreciate any resource based on any other axiomatisation, as I really can't find anything at all besides this brief mention by Thomas Jech.

Answer 1

I believe that you may have misstated the definition of what it means to be an $\in$-homomorphism. (I couldn't find your notion in Jech at your links — have I missed it?)

For example, with your definition, $f(X)$ is always a set, and never an atom, and so if there are atoms, then the identity function will not be a $\in$-homomorphism. Your definition seems to require us to map all atoms to $\emptyset$. Further, your notion would require $f$ to fix all well-founded sets, since there could be no $\in$-minimal set that is moved. In ZFC, therefore, your notion trivializes. Your definition also seems to presume that the models are transitive, using the standard $\in$-relation, whereas the notion of homomorphism should be sensible with arbitrary models of set theory.

I believe that the intended notion of homomorphism should be the one obtained by viewing models of set theory as relational structures $\langle M,\in^M\rangle$, a domain $M$ with a binary relation $\in^M$. In this case, the standard model-theoretic notion of (strong) $\in$-homomorphism or $\in$-embedding between two such structures would be a map $j:M\to N$ such that $$x\in^M y\iff j(x)\in^N j(y)$$ for all $x,y\in M$.

The difference between this and your notion is that you require that $j(y)$ has no other $\in^N$ elements except the $j(x)$ objects.

I analyzed this homomorphism notion in my paper

• Hamkins, Joel David, Every countable model of set theory embeds into its own constructible universe, J. Math. Log. 13, No. 2, Article ID 1350006, 27 p. (2013). ZBL1326.03046. Blog post

One of the main results was that the countable models of set theory are linearly pre-ordered by embedding:

Theorem. For any two countable models of set theory $\langle M,\in^M\rangle$, $\langle N,\in^N\rangle$, one of them is isomorphic to a submodel of the other.

This notion of submodel is the model-theoretic notion of $\in$-homomorphism, which are necessarily injective and thus isomorphisms of the domain with the range.

The theorem is often found surprising, but this is mainly because this notion of submodel is much weaker than we usually consider in set theory. In particular, submodels in this sense need not preserve much set-theoretic truth; they need not even be $\Delta_0$-elementary. This is a reason to view $\in$-homomorphisms as much weaker than we probably want. Set theorists typically want to consider emeddings that are at least $\Delta_0$-elementary, preserving $\Delta_0$ truth.

To illustrate the difference, the map $j:V\to V$ defined by $$j(y)=\{j(x)\mid x\in y\}\cup\{\{\emptyset,y\}\}$$ has $x\in y\iff j(x)\in j(y)$, and it is therefore an $\in$-embedding of $V$ into $V$. And yet, it has no fixed points — it does not even fix the natural numbers to themselves, and it does not carry $\emptyset$ to $\emptyset$.

Another part of my answer to your question is to mention my paper (updated from my initial post, which had mentioned the wrong paper):

• Daghighi, Ali Sadegh; Golshani, Mohammad; Hamkins, Joel David; Jeřábek, Emil, The foundation axiom and elementary self-embeddings of the universe, Geschke, Stefan (ed.) et al., Infinity, computability and metamathematics. Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch. London: College Publications (ISBN 978-1-84890-130-8/hbk). 89-112 (2014). ZBL1358.03070. Blog post

In that paper, we considered for various anti-foundational theories whether there can be nontrivial elementary embeddings $j:V\to V$. Some of the arguments involve Quine atoms in the anti-foundational theories, but these function essentially similarly to urelements in ZFA, and the relevant notion of embedding is your notion of $\in$-homomorphism. I'd encourage you to take a look there.

Answer 2

This answer has nothing to do with atoms, but on page 69 of Jech's book you'll find Mostowski's Collapsing Theorem. The map used there is exactly like the ones you mention. These collapsing functions are important tools for dealing with constructible sets, large cardinals, etc.