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>In particular, a set can contain itself.

There are many kinds of mathematics, this is an unusual one.

"In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself."

https://en.wikipedia.org/wiki/Universal_set



It is mentioned 2 sentences earlier that it was the case in naive set theory.


This gives rise to Russell's Paradox:

Does the set of "all sets that do not contain themselves" contain itself?


You can have set theories that allows sets to contain themselves without allowing Russell's Paradox. You can read about non-well-founded set theory[1] if you're curious.

[1]: https://en.wikipedia.org/wiki/Non-well-founded_set_theory


I like the diagram of "the set containing itself". It illustrates non-well-foundedness niceley.


Which diagram are you referring to?


This one:

https://abuseofnotation.github.io/category-theory-illustrate...


Resolving this paradox is discussed in TFA as being the founding rationale for the Zermelo–Fraenkel set theory in fact.


simple: there is no such set.




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