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I am interested in equivalent formulations of the extensionality axiom for the purpose of constructing non-classical models of set theory. I was wondering whether the following two principles are classically equivalent (over ZFC minus Extensionality)?

(Ext*) $\forall u \forall v [\forall w (u \in w \leftrightarrow v \in w )\rightarrow u=v]$

and

(Ext) $\forall u \forall v [\forall w (w \in u \leftrightarrow w \in v )\rightarrow u=v]$

Eddie Chau
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  • Classically equivalent over what base theory? They're certainly not equivalent "in a vacuum" - for example, in the single-sorted version of $\mathsf{NBG}$ one is provable and the other is disprovable (since proper classes are not elements of anything). – Noah Schweber Oct 18 '21 at 01:35
  • @NoahSchweber ZFC! – Eddie Chau Oct 18 '21 at 01:36
  • Actually, are you sure you've written these correctly? I think you want the "$\forall w$" inside the hypothesis on the inner conditional; otherwise, each is $\mathsf{(ZFC-Extensionality)}$-provably false (take $u=1,v=2,w=\emptyset$). – Noah Schweber Oct 18 '21 at 01:38
  • And even after that fix, they'll each be outright provable in $\mathsf{ZFC}$ hence boringly equivalent over $\mathsf{ZFC}$, so equivalence over $\mathsf{ZFC}$ doesn't seem like the right thing to look at here. Maybe you want equivalence over $\mathsf{ZFC-Extensionality}$ instead? – Noah Schweber Oct 18 '21 at 01:39
  • Yes, that's what I mean! Are they equivalent over ZFC minus extensionality? – Eddie Chau Oct 18 '21 at 01:42
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    I might be misreading some quantifier scopes. Is your question different from the following one: https://math.stackexchange.com/questions/3924887/does-equality-of-sets-follow-not-only-from-what-they-contain-but-also-from-what – Z. A. K. Oct 18 '21 at 01:44
  • Yes, that was exactly my question. However, do the theories ZFC and ZFC-minus extensionality + extensionality* not have the same collection of theorems? – Eddie Chau Oct 18 '21 at 01:50
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    No they don't. As per my answer to the previously linked question, ZFC-extensionality proves extensionality*. However, ZFC-extensionality does not prove extensionality itself. So extensionality is not a theorem of ZFC-ext+ext* either. – Z. A. K. Oct 18 '21 at 01:55

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