TITLE: Is axiom of choice necessary for proving that every infinite set has a countably infinite subset?
QUESTION [19 upvotes]: Is it possible to prove the following fact without axiom of choice ?
" Every infinite set has a countably infinite subset". 
Can it be proved that axiom of choice is necessary here ?

REPLY [3 votes]: In very short terms (you are allready overwhelmed by good answers of experts): 
$$\omega\leq_{1}A\implies\neg A<_{1}\omega$$ 
Or in words:$$A\text{ is Dedekind-infinite}\implies A\text{ is infinite}$$
But for the converse you need a weakened form of AC. 
The axiom that every countable set has a choice-function (denoted as CC) will do.
This axiom implies that: $$A\leq_{1}\omega\vee\omega\leq_{1}A$$