5 ms·
>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 regularit
by hackandthink 3y ago
>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 https://en.wikipedia.org/wiki/Universal_set
- axblount 3y agoThis gives rise to Russell's Paradox: Does the set of "all sets that do not contain themselves" contain itself?
- tikhonj 3y agoYou 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 https://en.wikipedia.org/wiki/Non-well-founded_set_theory
- hackandthink 3y agoI like the diagram of "the set containing itself". It illustrates non-well-foundedness niceley.
- l33t7332273 3y agoWhich diagram are you referring to?
- hackandthink 3y agoThis one: https://abuseofnotation.github.io/category-theory-illustrated/01_set/set_contains_itself.svg https://abuseofnotation.github.io/category-theory-illustrate...
- seanhunter 3y agoResolving this paradox is discussed in TFA as being the founding rationale for the Zermelo–Fraenkel set theory in fact.
- righttoolforjob 3y agosimple: there is no such set.
- bmacho 3y agoIt is mentioned 2 sentences earlier that it was the case in naive set theory.