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I'm trying to prove that if $X$ is dedekind finite then so is $X^X$ and I do not want to rely on the fact that dedekind finite is equivalent to finite when assuming AC (however we can assume AC). That is, I'm looking for a proof that demonstrates directly that if $m : X^X \to X^X$ and $m$ is a monomorphism then it is also an epimorphism assuming that $X$ is also dedekind finite. In other words, I'm seeking a proof that does not mention cardinals.

Also I have already reviewed these two posts:

(1) https://mathoverflow.net/questions/179434/exponentiation-and-dedekind-finite-cardinals

(2) Example of a set of real numbers that is Dedekind-finite but not finite

Eric Wofsey
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Squirtle
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    The best I can think of is that if $A$ is an infinite Dedekind-finite set, then either $\mathcal P(A)$ is Dedekind-infinite or $\mathcal{P(P(}A))$ is Dedekind-infinite. In either case, it shows that Dedekind-finite cardinals are not closed under exponentiation. So, incorporate the proof that Dedekind-finite cardinals are finite into the above, and you're good to go. But it's definitely not a "direct" proof in the sense that it really just an implementation of Greenspun's tenth rule, applied to cardinals. – Asaf Karagila Jan 16 '22 at 01:06
  • hi @AsafKaragila. how do you prove that, if $A$ is infinite Dedekind-finite, then one of $P(A)$ and $P(P(A))$ is Dedekind-infinite? – Atticus Stonestrom Dec 03 '23 at 01:58
  • @Atticus: If $A$ is any set, then $A$ surjects onto $\omega$ if and only if $\mathcal P(A)$ is Dedekind infinite. And if $A$ is any set, then $A$ is infinite if and only if $\mathcal P(A)$ surjects onto $\omega$. – Asaf Karagila Dec 03 '23 at 08:42
  • @AsafKaragila thanks! – Atticus Stonestrom Dec 03 '23 at 13:15
  • I forgot about this post, but I am still interested in it. The point was that Dedekind-finite = finite in the setting I am working in, I just want to avoid reference to cardinals. So it's not clear to me how there could be an infinite Dedekind-finite set when we assume the AC. – Squirtle Feb 02 '24 at 17:24

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