4 ms·
> 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 referri
by dkbrk 3y ago
> 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/ https://plato.stanford.edu/entries/nonwellfounded-set-theory...
[1]: https://en.wikipedia.org/wiki/Aczel%27s_anti-foundation_axiom https://en.wikipedia.org/wiki/Aczel%27s_anti-foundation_axio...
- daxfohl 3y agoMy understanding was that you just had to get rid of the law of the excluded middle. Then things like Russell's Paradox can be stated, but evaluate to neither true nor false, and be fine. Is that not correct?
- climatologist 3y agoIt's more to do with unrestricted circular self-reference than excluded middle but what you're saying makes sense. Without excluded middle there is no contradiction. There is something called Curry's paradox which is similar. Excluded middle is not involved in Curry's paradox but circular self-refernece is still the culprit for the paradoxical conclusion.