Be careful, there are such things as transfinite numbers, and they are well-founded. They let us do wonderful things like transfinite induction, and to prove, for example, that Goodstein's Theorem is true, even though it's unproveable in Peano Arithmetic.

But transfinite numbers are not "natural numbers", they are not in the set N, they don't have infinitely many digits, and in the context of this thread, they are a red herring.