# Dedekind-infinite vs weakly Dedekind-infinite in ZF

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.

• "$S$ 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. Commented Apr 15 at 7:06

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.
• 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. Commented Apr 15 at 9:58