Is natural numbers set $\mathbb N$ infinite set? A set with uncountable number of elements is called an infinite set.
Is that the set of all natural numbers, $\Bbb N=\text{{$1,2,3,\ldots$}}$ infinite set?
As far i know $\Bbb N$ is "countably" infinite or denumerable.
That is, i can't say $\Bbb N$ is "uncountable".
Then how  is $\Bbb N$ an infinite set?

Finite set is a set that for some natural number $k$, there is a bijection between the set and $\{1,\ldots,k\}$.
An infinite set is a set which is not finite. $\Bbb N$ is not finite, so it is infinite.
Countably infinite set is a set that has a bijection between itself and $\Bbb N$. An uncountable set is a[n infinite] set which is not countable. So an infinite set can be countably infinite or uncountable.
(If it bothers you that the definition of finiteness appeals to the natural numbers there are set theoretical definitions which makes no such appeal. For example Tarski's definition $A$ is finite if and only if every non-empty $U\subseteq\mathcal P(A)$ has a $\subseteq$-minimal element; or Dedekind's definition $A$ is finite if and only if every injection from $A$ to itself is a bijection.
Interestingly the equivalence between the definition in my first line, and Tarski's requires no use of the axiom of choice, whereas the equivalence with Dedekind's definition does require us to use the axiom of choice.)
