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I have some concerns over the definitions of Dedekind-infinite and weakly Dedekind-infinite as shown in the wikipedia page https://en.wikipedia.org/wiki/Dedekind-infinite_set

I suspect there is a mistake in the wiki, but I would like to get some clarification on the matter first.

From the wiki, Dedekind-infinite refers to any non-empty set $S$ such that there is a bijection between $S$ and some proper subset of $S$. It can be shown (within ZF) that this is equivalent to there existing an injection from $\mathbb{N}$ to $S$, which in turn equates to there existing a surjection from $S$ to $\mathbb{N}$. Great so far.

Now let's look at the definition of weakly Dedekind-infinite, which is that there exists a surjection from $S$ onto some countably infinite set. This doesn't seem correct, because this condition almost trivially equates to there existing a surjection from $S$ to $\mathbb{N}$. If the definition is indeed correct, then weakly Dedekind-infinite would be the exact same thing as Dedekind-infinite, which seems odd. As far as I know, the two terms are equivalent if and only if some form of the axiom of choice is true, but within ZF alone they are not equivalent. So something seems awry.

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    $\begingroup$ "$S$ is Dedekind-infinite" is not equivalent (in ZF) to "there is a surjection from $S$ to $\mathbb N$" and the Wikipedia page you linked to does not say that it is. $\endgroup$
    – user14111
    Commented Apr 15 at 7:06

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So apparently there is no mistake on the wiki, but rather I have implicitly used the "fact" that an injection from $S$ to $T$ exists if and only if a surjection from $T$ to $S$ exists. I just learned that this claim is not true within ZF alone but is true within ZFC.

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  • $\begingroup$ Indeed! Strictly speaking you want to talk about "a surjection from $T$ to $S$ exists or $S$ is empty", since there is no surjection from a non-empty set to the empty set. In ZFC, this is equivalent to "an injection from $S$ to $T$ exists". But in fact, the theorem stating this equivalence is itself equivalent to the axiom of choice, which you may like to think about. $\endgroup$ Commented Apr 15 at 9:58
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    $\begingroup$ @IzaakvanDongen That’s the partition principle and it’s an open question whether it’s equivalent to AC. $\endgroup$ Commented Apr 15 at 20:36
  • $\begingroup$ @spaceisdarkgreen, you're absolutely right, thank you for the correction. That was silly of me! (I got mixed up with "every surjection has a right inverse"...) $\endgroup$ Commented Apr 15 at 20:40

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