Defining "Countably Infinite"

Question

I was reading about countably infinite sets and the definition goes as, "A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers"(Source: Google). Well yes, this make sense. As the set $\mathbb{N}$ is countably infinite and set which can it can create a bijection with, would end up having the same number of elements and thus they too would be countably infinite. But this would be true iff $\mathbb{N}$ is countably infinite. How would you prove that? Or is the idea of countably infinite base on some assumption that $\mathbb{N}$ is countably infinite?

Answer 1

We don't assume that $\Bbb N$ is countably infinite. We just name sets which can be put in bijection with $\Bbb N$ as countably infinite.

Then, it is a trivial consequence that $\Bbb N$ is countably infinite, as the identity function witnesses this fact.

Answer 2

That $N$ is countably infinite is not so much an assumption as it is a basic intuition. When we say a collection is countable, we are saying we can pair its items off with successive elements of $N$ because intuitively that is what counting actually is - a pairing off. And of course $N$ is infinite because intuitively, no matter how high we count, we can always count one more. So intuitively, something is countably infinite if it can be counted using all of $N$ and nothing less.

When we get to Axiomatic Set Theory, this intuition is formally embedded into the theory by postulating the Axiom of Infinity. So within the theory we can define 'countably infinite' as 'pairable with all elements of the set provided by the Axiom of Infinity'.

[so actually, your final question essentially gets to the heart of the matter]