Q:

Is $...000n$ not a natural number?

Is $...000n$ (i.e a natural number with an infinite number of zeros on the left) a natural number?
If it is a natural number, can't we use something like Cantor's diagonalization method to prove that $\mathbb{N}$ is uncountable (I know that it is false, and that $\mathbb{N}$ is countable)?
Thanks in advance

A:

To answer the other question: Cantor diagonalization cannot be used in this case: if you try using it for "most" choices, the digit you'll have will be $0$. So infinitely many times, you'll need to pick a digit $a \neq 0$, but this would create an expression with infinitely many non-zero digits, which is not a number.

