From Wikipedia:
The axiom of regularity is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A.
I'm sure I'm grossly misunderstanding this, but it doesn't seem to make any sense. Two disjoint sets do not have elements in common, correct? So this says a set contains an element that it is disjoint with. So wouldn't this mean that the set doesn't contain that element if it is disjoint with it? To me, this axiom seems to be saying "Every non-empty set contains an element that it doesn't contain." This obviously doesn't make sense, so I'm definitely misunderstanding at least one aspect of this axiom, but which part?
Best Answer
As an example, let $X = \{1,2\}$ (where $0=\varnothing, 1= \{0\}, 2=\{0,1\}$). Then $$X\cap 1 = \varnothing$$ $$X\cap 2 = \{1\}$$ Due to the first equality, our set satisfies the axiom of regularity. $X$ being disjoint with $1$ does not imply that $X$ does not contain $1$; $X\cap 1 = \varnothing$ is a completely different statement from $1\notin X$. Indeed, the former holds, as the only element of $1$ is $0$, which is not an element of $X$.