Like what the title said, I'm interested to know which of the following is provable in ZF. As much as possible I will give my thoughts and reasoning for each choice.
1.) If $\kappa$ is an infinite cardinality then $\aleph_0\leq \kappa$
2.) If $\kappa$ is an infinite cardinal number then $\aleph_0\leq \kappa$
3.) If $\kappa$ is an infinite cardinality, then $\kappa \cdot\kappa = \kappa$
For 3.) I just followed my gut feeling and say it is not provable without choice since it involves some form of cardinal arithmetics, though I do wish for a more formal reason.
My struggle is between 1. and 2.
For 1.) If $\kappa$ is an infinite cardinality, then it means there is an injection from $\omega = \aleph_0$ into $\kappa$, hence the inequality. This sounds right.
For 2.) While it is true we need Choice to assign cardinal number to a infinite set $A$, but if we already know $\kappa$ is some $\aleph$'s, then it also makes sense for the inequality to hold.
So I really wish to know where in my reasoning is wrong, and an explanation for the right options.Of course further insights on similar question is more than welcome.