TITLE: Applications of the Lawvere Fixed Point Theorem for Sets
QUESTION [5 upvotes]: I'm not familiar with the general theorem for closed, cartesian categories (as I'm not familiar with closed, cartesian categories), but I am aware of this version of the fixed point theorem for sets:

Let $A, B$ be sets. If there exists a surjective function $f: A \longrightarrow \textbf{Set}(A,B)$, then every function $g:B \longrightarrow B$ has a fixed point.

I'm aware of two applications of the theorem: proving Cantor's theorem (setting $B=\{0,1\}$) and proving that $[0,1]$ is uncountable (setting $A=\mathbb{N}$, $B= \{0,1\}$, looking at binary representations).
Are there any other neat applications of this fixed point theorem for sets? For instance, can we deduce the Tarski fixed point theorem for sets (every non-decreasing endofunction of a power set of a set has a fixed point) or the Cantor-Bernstein theorem using this theorem?
Just thought this result was cool, and wanted to see what other things you could do with it. Thanks in advance for any replies!

REPLY [3 votes]: The Lawvere fixed point theorem has limited applications in $\text{Set}$ because the only set with the fixed point property is the one-element set $1$, so if $B$ is any other set you just conclude that there can't be a surjection $A \to [A, B]$, which for $B \neq 0, 1$ is basically Cantor's theorem. 
It has more interesting applications in other cartesian closed categories precisely because sometimes these categories have more interesting objects with the fixed point property. For example, there are some cartesian closed categories consisting of a single object $X$; in particular, this means that $X \cong [X, X]$, and applying the Lawvere fixed point theorem to an isomorphism of this form allows you to conclude that $X$ has the fixed point property. CCCs like this model the untyped lambda calculus, and writing down the fixed points explicitly gives you the Y combinator.