Proposition: For a set $X$ and its power set $P(X)$, any function $f\colon P(X)\to X$ has at least two sets $A\neq B\subseteq X$ such that $f(A)=f(B)$.
I can see how this would be true if $X$ is a finite set, since $|P(X)|\gt |X|$, so by the pigeonhole principle, at least two of the elements in $P(X)$ would have to map to the same element.
Does this proposition still hold for $X$ an infinite set? And if so, how does this show that the set of all singleton sets cannot exist?
Best Answer
If $x$ is a set containing all singletons, $\cup x$ is a set containing all sets, which leads to Russell's paradox. Thus, in $ZF$ there is no such $x$.