myren

"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.

I don´t think this is an alien that applied downwards Löwenheim-Skolem to the reals. Rather I think this is an alien which applied upwards Löwenheim-Skolem to peano arithmetic. I think this because the alien talks about infinite natural numbers, aka non-standard natural numbers. It has used upwards Löwenheim-Skolem to get a model of peano arithmetic that has the cardinality of the continuum, and then there is of course a bijection between our reals and the thing that it calls the natural numbers. I see no sign that it´s using a non-standard model of the reals, but lots of signs that it´s using a non-standard model of the naturals.