My question is the following, if $\kappa$ is an aleph and $F$ is the set of all finite sequences in $\kappa$, then the fact that $|F|=\kappa$ is provable from $ZF$?.
This can be proven from $ZF$ for the case $\kappa=\aleph_0$ using the fundamental theorem of arithmetic, the fact that there exist infinitely many primes and that $\kappa\cdot\kappa=\kappa$; all facts provable in $ZF$. It can be proven also for the case when $\kappa$ is an epsilon number using the uniqueness in the Cantor's Normal Form Theorem.
But I don't know how to begin a proof for the general case.
So I want to know if this is true.