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I was reading Terence Tao's text on analysis. After stating the axioms of pairing and regularity, he asks for proof of the statement that no set can be an element of itself, using the above two axioms. He has not defined any concepts like hierarchy or ranks.

I can see how,

A [itex]\notin[/itex] A

ifA={A}, from the axiom of regularity.

But if theAwere to contain itself any other seta, such thatA={a, A}, where saya={1}, then we would have

a [itex]\cap[/itex] A ≠ ∅

SinceAcontains the setaand not the elements ofa, the intersection would be disjoint. Alternately, if the setAwere defined asA={1, A}, the intersection

1 [itex]\cap[/itex] A ≠ ∅

Isn't it possible to have such sets? What role does the axiom of pairing play in the preventing the existence of such sets?

Thanks.

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# Regularity and self containing sets

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