A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. In other words, one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.

What does countably infinite mean?

Any set which can be put in a one-to-one correspondence with the natural numbers (or integers) so that a prescription can be given for identifying its members one at a time is called a countably infinite (or denumerably infinite) set.

What is uncountable infinity called?

Without the axiom of choice

(namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above, but not the fourth characterization.