In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way:
It went as follows. Let T be a theory with a nonstandard model. By virtue of his Incompleteness Theorem, the consistency proof of T cannot be carried out within T. Consequently, T and the proposition "T is inconsistent" is consistent. There is, therefore, a natural number N which is the Gödel number of a proof leading to a contradiction from T. Such a number is obviously an infinite natural number.
Can someone elaborate on the final sentence of this quote?
(Also, in what sense is this teaching nonstandard models if it assumes them from the start?)