TITLE: Is it consistent to have a function that is sensitive to subset relation from the power set of a set to that set?
QUESTION [9 upvotes]: Is it consistent with $ZF$ to have a set $S$ and a function $F: P(S) \to S$ such that:

$\forall X,Y \in P(S): X \subsetneq Y \implies F(X) \neq F(Y)$

REPLY [8 votes]: Here's an argument that doesn't use ordinals, as an alternative to the nice proof described by Andrés and Andreas.
Take $F: P(S) \to S$ satisfying your hypothesis. Define a function $\Phi: P(S) \to P(S)$ by
$$
\Phi(X) = \{F(Y): Y \subseteq X\}.
$$
Then $\Phi$ is an order-preserving map from the complete lattice $P(S)$ to itself, so there is a least $X \in P(S)$ such that $\Phi(X) \subseteq X$. (That such an $X$ exists is part of the proof of the Knaster-Tarski fixed point theorem, and is in any case easy: put $X = \bigcap\{ Y \in P(S): \Phi(Y) \subseteq Y\}$, then use the fact that $\Phi$ is order-preserving to show that $\Phi(X) \subseteq X$.)
Now:

$F(X) \in \Phi(X)$ by definition of $\Phi$, so $F(X) \in X$, so if we write $X' = X \setminus\{F(X)\}$ then $X' \subsetneqq X$.

$\Phi(X') \subseteq \Phi(X) \subseteq X$, so $\Phi(X') \subseteq X$.

Any subset $Y$ of $X'$ is a proper subset of $X$, so your hypothesis on $F$ gives $F(Y) \neq F(X)$.  Hence $F(X) \not\in \Phi(X')$.


Combining the three bullet points, we have a proper subset $X'$ of $X$ satisfying  $\Phi(X') \subseteq X'$. This contradicts the minimality of $X$.