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I've already found some similar questions in here (and other sites), but in most of the case, the use of axiom of foundation is required to complete the proof. Is there any way to prove $\not\exists x(x\in x)$, without using the axiom?

Thank you in advance.

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See this comment. –  Git Gud Apr 20 at 12:44
2  
Here are some lecture notes by Aczel on non-well-founded set theory. –  MJD Apr 20 at 13:04
    
The set of sets with more than one element contains itself... Right? –  BlackBear Apr 20 at 19:07
    
@BlackBear In ZF set theory, there is no such set. If Morse-Kelley set theory, there is a class of all sets with more than one element, but this class is provably not a set. I think the situation in NFU set theory is similar to MK. –  MJD Apr 29 at 13:30

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up vote 12 down vote accepted

You can't.

It is consistent that the axiom of foundation fails, and there are sets of the form $x=\{x\}$. More generally there are "Anti-Foundation Axioms" which assert the existence of ill-founded sets. The more famous ones are by Boffa and by Azcel.

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Uh, where exactly? –  Asaf Karagila Apr 20 at 12:49
1  
Nevermind, I was parsing it wrongly. –  Git Gud Apr 20 at 12:50
    
Okay then. Because I've been up an ungodly number of hours, and it's very feasible that I made a typo. –  Asaf Karagila Apr 20 at 12:52

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