> A set cannot be part of itself in an axiomatic formulation of set theory.

This isn't quite true. In practice, you're correct: "set theory", generally referring to ZF (or some closely related derivative thereof) with the Von Neumann Universe, doesn't allow sets to contain themselves. But it is possible to axiomatize set theory where sets can contain themselves [0].

This replaces the axiom of foundation with the axiom of anti-foundation [1], so it's not naive set theory but it is an axiomatized non-well-founded set theory.

[0]: https://plato.stanford.edu/entries/nonwellfounded-set-theory...

[1]: https://en.wikipedia.org/wiki/Aczel%27s_anti-foundation_axio...