The set of natural numbers is infinite and countable. Ok. But think of an object with infinite digits (141258173412873....). Is it a natural number?
Edit: What i found confusing was the fact that, since $\mathbb{N}$ is an infinite set, an object with infinite digits should be also a number and should belong to $\mathbb{N}$. I know this is a naive view. But now things are clearer to me, thanks to your answers! If i had to explain to a person not (too) educated in mathematic what $\mathbb{N}$ (the set of natural numbers) is, i would start with this:
consider the following algorithm (procedure) to construct $\mathbb{N}$={1,2,3,4....}:
- num = 1
- $\mathbb{N}$ is the empty starting set of numbers
- put num in $\mathbb{N}$
- num = num + 1
- repeat from 3
Now, does $\mathbb{N}$ has objects with an infinite number of digits? No. The procedure goes on forever, but everytime we add a number to $\mathbb{N}$ (step 3), the number we are adding has a finite number of digits.
This view is only slightly different from other answers given to my original question, but i think it is simple enough to explain why a procedure that goes forever and build objects with an increasing number of digits does not produce a set with objects with an infinite number of digits.