Why does the set of all singleton sets not exist?

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?

Solution 1:

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 $x$.

Solution 2:

If you had a function $f\colon P(X)\to X$ such that $f(A)=f(B)\Rightarrow A=B$, then $f$ would be one-to-one, which would imply $|P(X)|\leq |X|$; since $|X|\lt |P(X)|$ holds by Cantor's Theorem, that would give you $|P(X)|\lt|P(X)|$, which is impossible. This argument does not use finiteness of $X$, so it holds for $X$ infinite as well.

Suppose that the collection of all singletons is a set $X$. Define a map $f\colon P(X)\to X$ by $f(A) = \{A\}$; this is well defined, since $\{A\}$ is a singleton. If $A\neq B$, then $\{A\}\neq\{B\}$, so $f$ would be one-to-one. Since, by the proposition, this is impossible, we conclude that our assumption that the collection of all singletons is a set must be false. Thus, there is no "set of all singletons."

Solution 3:

If you accept that the collection of all sets (denoted by $V$ usually) is not a set (but in fact a proper class) then if you look at the function $f(x) = \{ x \}$ it is 1-1 from $V$ into the sets of all singletons, thus resulting that $V$ would be a set, in contradiction to the fact that it is in fact a proper class.

But of course you'd have to accept that $V$ is a proper class as well, this is simpler and follows by the argument given in Arturo's answer (that the power-set of $V$ would be of the same cardinality).

Solution 4:

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.)

Solution 5:

The question in the title is a consequence of the axiom of regularity. In fact If there was such a set $X$, then $\{X\}\in X$. But obviously we always have $X\in\{X\}$. Hence, we would be able to construct an infinite downward chain of sets which contradicts the axiom of regularity.