For example, the axiom of pairing says:
Let $a$ be a set.
Let $b$ be a set.
If follows that the set $\{a,b\}$ exists.
This can be used to prove the existence of singletons, for instance, by setting $b := a$ (in the previous statement). Namely, the axiom of pairing implies the following:
Let $a$ be a set.
If follows that the set $\{a\}$ exists.
This got me thinking. What ZFC axiom implies that, for any set $a$, the set $\{a,a\}$ equals the set $\{a\}$? Equivalently, what axiom of ZFC implies that the sets of ZFC don't behave like multisets? (I suspect it's extensionality, but I couldn't argue why. So, if it is extensionality, then I'm gonna need some convincing...)
from Hot Weekly Questions - Mathematics Stack Exchange
étale-cohomology
Post a Comment