Is a paradoxical partition of a set smaller than the power set of that set? When it is said that almost every model of $\sf ZF + \neg AC$ do have paradoxical partitioning $\sf PP$, that is: there exists a set $X$ and a partition $P$ of $X$ that is strictly larger than $X$. Is it also provable that: $$|P|<|\mathcal P(X)| \text{ ?}$$ 
 A: Yes.  Note that trivially $|P|\leq|\mathcal{P}(X)|$ since $P\subseteq\mathcal{P}(X)$.  If $|P|=|\mathcal{P}(X)|$, then we can define a surjection $X\to\mathcal{P}(X)$ by mapping each element of $X$ to the element of $P$ that contains it and then composing with a bijection $P\to\mathcal{P}(X)$.  This is impossible by Cantor's theorem.
A: Yes. If $P$ is a partition of $X$, then
$$|P|\lt|\mathcal P(P)|\le|\mathcal P(X)|$$
by virtue of Cantor's theorem and the obvious injection $\mathcal P(P)\to\mathcal P(X)$.
A: We can say a bit more.
Note that if there is a surjection $f:A\to B$ then $f^{-1}:\mathcal P(B)\to\mathcal P(A)$ is an injection. Also, if $g:C\to D$ is an injection and $C\ne\emptyset$, then mapping $D\smallsetminus g[C]$ to a fixed point of $C$ and everything else to their preimage by $g$ is a surjection from $D$ to $C$.
It follows that if $\sim$ is an equivalence relation on a set $X$ and $|X/{\sim}|>|X|$, then there are surjections from $X$ onto $X/{\sim}$ and also from $X/{\sim}$ onto $X$, and therefore there are injections from $\mathcal P(X)$ into $\mathcal P(X/{\sim})$ and vice versa. But then $|\mathcal P(X)|=|\mathcal P(X/{\sim})|$.
