Mathematics Asked by lockedscope on February 23, 2021

Axiom of Regularity/Foundation

∀? (? ≠ ∅ → ∃? (? ∈ ? ∧ ? ⋂ ? = ∅))

This says that every non-empty set must contain a set that shares no

elements with it.

So, how do we understand axiom of regularity is talking about a set with "y" just from the notation?

(I think using intersection ⋂ gives a clue that y is a set because only sets intersects not elements but not sure if its right inference.)

There's less here than meets the eye. In $mathsf{ZFC}$, **everything** is a set. So $y$ is a set because it can't not be.

(This is the source of the most common criticism of $mathsf{ZFC}$, incidentally: that formalizing mathematics in $mathsf{ZFC}$ inevitably leads to "junk theorems." See here for example.)

Correct answer by Noah Schweber on February 23, 2021

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