In mathematics, the axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo–Fraenkel set theory and was introduced by von Neumann (1925); it was adopted in a formulation closer to the one found in contemporary textbooks by Zermelo (1930). In first-order logic the axiom reads:
\(A \neq 0 \rightarrow (\exists x)[x \in A\ \&\ (\forall y)(y \in x\ \&\ y \notin A)]\)
In introducing this Axiom, Patrick Suppes says:
It is difficult to think of a set which might reasonably be regarded as a member of itself. Certainly the set of all men, for example, is not a man and is therefore not a member of itself. Perhaps it might be argued that in intuitive set theory the set of all abstract objects or the set of all sets should provide an example of a set which is a member of itself, but as we saw in the first chapter, the set of all sets is itself a paradoxical concept.
These remarks suggest we take as an axiom:
\(A \notin A\)
However, [this] assumption would not prohibit the counterintuitive situation of there being distinct sets \(A\) and \(B\) such that
\(A \in B\ \&\ B \in A\)
(If you do not believe [this] is counterintuitive, try to give a simple example of sets \(A\) and \(B\) satisfying [this]).
It’s funny that he would say that, because at that very instant I remembered that a recurrent technique for romantic music composers is to say that people “belong to each other”. Literally Lenny Kravitz writes
I belong to you, and you, you belong to me too.
The 88 also writes something similar in the lyrics of you belong to me:
you belong to me and I belong to you.
Other examples include the song I belong to you by Anastacia and I’m sure there has to be like over a dozen of these examples.
I wonder what would Patrick Suppes say…