# Without foundation: Not transitive model of ZFC?

Let's assume that ZFC is consistent.

It is easy to show that there is an illfounded model (with compactness, e.g. here). If we have the axiom of foundation at the background level we can conclude that such a model is not isomorphic to a transitive model.

Is it possible to show that there is a model of ZFC that is not isomorphic to a transitive model without assuming foundation at the background level?

• That's a nice question. Commented Nov 28, 2018 at 18:11

Nice question! The answer is no: Foundation is indeed needed in the metatheory.

Boffa's antifoundation axiom implies that every extensional directed graph $$(X; \rightarrow)$$ is isomorphic to $$(A;\in\upharpoonright A)$$ for some transitive set $$A$$. In particular, any model of ZFC is isomorphic to a transitive model - at least, that's what ZFC-Foundation+Boffa thinks.

Note that the only thing being used here is that ZFC is extensional, so the above also applies to e.g. Quine's theory NF and its variants. Even for non-extensional theories, we get a version of the above result: we get that if $$T$$ is a consistent $$\{\in\}$$-theory then $$T$$ has a model of the form $$(A;\in\upharpoonright A)$$ for some set $$A$$.

For a discussion of Boffa's axiom, see Chapter $$5$$ of Aczel's book.

• You mean ZFC-Foundation+Boffa proves "If ZFC has a model, then every model is isomorphic to a transitive model.", right? Commented Nov 28, 2018 at 19:07
• And do you know if there is a (finitistic) proof of Con(ZFC-Foundation) $\Rightarrow$ Con(ZFC-Foundation+Boffa)? Commented Nov 28, 2018 at 19:13
• @PopovFlorino "If ZFC has a model, then every model is isomorphic to a transitive model." Yes (although what I wrote is also true, it's just weaker). As to relative consistency, I believe that it does - the point being that any model $M$ of ZFC interprets a model of ZFC-Foundation+Boffa consisting more-or-less of the extensional graphs in $M$. Commented Nov 28, 2018 at 19:21
• I think you should state this in a stronger form: Any model is isomorphic to a transitive model. I am quite surprised this holds. Commented Nov 28, 2018 at 19:27
• @PopovFlorino Not a ton, sadly! It's not as popular as Aczel's antifoundation axiom. I think the Holmes-Forster-Libert article "Alternative set theories" in the Handbook of the History of Logic (vol. 6) is a good starting place. Commented Nov 28, 2018 at 19:47