TITLE: Why does the set of all singleton sets not exist?
QUESTION [21 upvotes]: 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?

REPLY [3 votes]: Your presupposition is incorrect; there are a number of set theories (some of them provably equiconsistent with ZF) in which the set of all singletons exists; see Limiting set theory using symmetry, or Forster’s Oxford Logic Guide on the subject.  (Disclaimer:  Forster discusses my work in the book, so I’m hardly unbiased, but it is the standard work.)