>>12514889There's often graph models to interpret the set theory. E.g. the class of Hereditarily finite sets correspond biject exactly with the rooted trees that have no non-trivial automorphism. I'ma post two very different such models (example picture). Sets containing themselves "just" correspond to different directed graph models. I suppose if one is willing to deal with the power set of the powerset of the naturals, one can also deal with sets of infinite depth such as x={x}. Those structures find some applications in logic/CS topics.
https://en.wikipedia.org/wiki/Non-well-founded_set_theoryThis was done in large parts by Aczel, which incidentally
>>12514951 also mentions
I don't want to get into your guys discussion, but I'd rather characterize sets as reified properties, i.e. properties which can exhibit properties themselves (you can say class instead of property), and such that the always-false property is a set.
Virtually always, the axioms about conjunction, disjunction and a notion of downward flattening of the properties are adopted (Empty Set, Pairing, Separation and Union, introducing algebraic manipulation rules to the game)
>>12514936>WHAT DO YOU THINK OF UR_ELEMENTSI think they make things more complicated and I assume they were mostly introduced by people who genuinely took set theory as a good foundation. I never used them.
