"If there really are an infinite number of natural numbers then some of them must be of a transfinite number of digits or else you would be including numbers in the list more than once."

Theorem: There are no infinite natural numbers.

Proof: By Induction:

Induction Start: 0 is finite.

Induction Step: If n is finite then n+1 is finite.

The principle of induction then tells us: All natural numbers are finite.

If all natural numbers are finite, then there are no infinite natural numbers.

End of proof.