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We have the AOE as:

$\forall A\,\forall B\,(\forall X\,(X\in A\iff X\in B)\Rightarrow A=B)$

and possibly its converse? I am not sure of the technical details.

I am confused as to why this implies equality, instead of defining it. Currently, I am reading through Naive Set Theory by Halmos. So far, in this set theoretic 'universe' we are constructing, we only have the notion of belonging ($\in$) and a set as a collection of 'things'. So, how can we bring in $=$ without defining it?

ngc1300
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  • Equality is usually considered a basic part of the logic, and not something that needs to be added in the theory. – Malice Vidrine Dec 13 '18 at 17:57
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    Note that if equality isn't part of your underlying logic, two sets being coextensive is not a good notion of equality; you could have $A$ and $B$ being coextensive, but $B\notin{A}$. If you want to define equality, you need an additional axiom ensuring that $A$ and $B$ being coextensive implies that they are also members of exactly the same sets. – Malice Vidrine Dec 13 '18 at 18:02
  • Can you provide and example? – ngc1300 Dec 13 '18 at 18:12
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    @pmac An example of what? – Noah Schweber Dec 13 '18 at 18:53
  • An example of two sets $A, B$, that are coextensive but $B \not\in { A }$. – ngc1300 Dec 13 '18 at 21:25

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