A strong Cantor theorem without choice

Question

Let $X$ be an infinite set. Can one prove in ZF (without choice, and with finiteness defined as being equipotent to a finite ordinal) that the set $\mathfrak{P}_{< \omega}(X)$ of all finite subsets of $X$ cannot be mapped onto the set $\mathfrak{P}(X)$ of all subsets of $X$?

If not, does countable choice suffice?

Answer 1

It is provable that $\mathcal P_\omega(X)$ is strictly smaller than $\mathcal P(X)$, but it is consistent that there is a surjection still.

Lorenz Halbeisen and Saharon Shelah, Consequences of arithmetic for set theory, J. Symbolic Logic 59 (1994), no. 1, 30--40.