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Tyrrell_McAllister comments on But Somebody Would Have Noticed - Less Wrong
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If you are talking about things that are set-like, except that they don't satisfy the extensionality axiom, then you just aren't talking about sets. The things you're talking about may be set-like in some respect, but they aren't sets.
There are other set-like things that don't satisfy extensionality. For example, two different properties or predicates might have the same extension.
To be clear - Aczel's ZFA and similar systems do satisfy extensionality; they'd hardly be set theories if they didn't. It's just that when you have sets A and B such that A={A} and B={B}, you're going to need stronger tools than extensionality to determine whether they are equal.
Interesting. I'm not familiar with Aczel's system. But is that what cousin_it is talking about doing? That looks like an adjustment to Foundation rather than to Extensionality.
It's both at once. (Though, as I said, you don't throw out extensionality. Actually, that raises an interesting question - could you discard extensionality as an axiom, and just derive it from AFA? I hadn't considered that possibility. Edit: You probably could, there's no obvious reason why you couldn't, but I honestly don't feel like checking the details...)
If you just throw out foundation without putting in anything to replace it, you have the possibility of ill-founded sets, but no way to actually construct any. But the thing is, if all you do is say "Non-well-founded sets exist!" without giving any way to actually work with them, then, well, that's not very helpful either. Hence any antifoundational replacement for foundation is going to have to strengthen extensionality if you want the result to be something you want to work with at all.
I think you mean to say is "non-Well-founded sets exist!" since you are talking about the antifoundational case (and even with strong anti-foundation axioms I still have well-founded sets to play with also).
Oops. Fixed.