# If $X=\bigcup_{n\in{\mathbb{N}}}\kappa^n$, is it provable from $ZF$ that $|X|=\kappa$?

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.

Thanks

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SKETCH: The Gödel pairing function gives you an explicit bijection $\varphi:\kappa\times\kappa\to\kappa$, and recursion over $\omega$ gives you explicit bijections $\psi_n:\kappa^n\to\kappa$ for $n\in\omega\setminus 1$. The map $$(\omega\setminus1)\times\kappa\to\kappa:\langle n,\alpha\rangle\mapsto\varphi\big(n,\psi_n^{-1}(\alpha)\big)$$ is then an injection.
Added: You may also find helpful the last part of this answer by Andres Caicedo to a question on MathOverflow; it gives in a little more detail a different argument that there is in ZF a class function that assigns to each infinite ordinal $\alpha$ a bijection between $\alpha\times\alpha$ and $\alpha$, which is all that you need to justify the map above.