Equivalent Conditions for Dedekind-Infinite Set

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Theorem

For a set $S$, the following conditions are equivalent:

$\left({1}\right):\quad$ $S$ is Dedekind-infinite.
$\left({2}\right):\quad$ $S$ has a countably infinite subset.

The above equivalence can be proven in Zermelo-Fraenkel set theory.

If the axiom of countable choice is accepted, then it can be proven that the following condition is also equivalent to the above two:

$\left({3}\right):\quad$ $S$ is infinite.


Proof