Lenny Kravitz and the Axiom of Regularity

Posted in: Math , Philosophy , Fun Fact

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…