Many of the states of affairs about infinite cardinalities and their size, depend on the axiom of choice. What would a comprehensive list be of the properties of the cardinality of the natural numbers N, be if we do not allow anything beyond ZF set theory? Surprise me.
Another question would be about the axiom of countable choice? Why do people accept this but not the axiom of choice proper? It is not intuitive (only less counterintuitive). Yes you can choose an element one by one, but this "intuition" leads to an inductive proof of finite choice for arbitrarily large finite n, not countable choice...
Is it accepted more sheerly on a practical basis? Because a theoretical one seems nonexistent to me.
Edit: I realised this is two questions, but perhaps countable choice is at least a connection between the two.