Gödel's way of teaching non-standard models to Takeuti.

About the final sentence: If the Goedel number, in some model, of a proof of a contradiction were a standard natural number, then it would encode an actual proof of a contradiction. But there is no actual proof of a contradiction in $T$ because, by hypothesis, $T$ has a model.

About assuming that $T$ has a nonstandard model: That assumption could be replaced with "$T$ has a model" and the argument would still work. So I conjecture that "nonstandard" in the first sentence was just a mistake.

This argument applies only to theories $T$ that are recursively (or at least definably-in-$T$) axiomatizable, because one needs to apply the incompleteness theorem.

Finally, in nonstandard analysis, one usually uses "nonstandard model" to mean an elementary extension of the standard model. That isn't what this argument produces, though. It produces a model satisfying "$T$ is inconsistent" whereas the standard model satisfies "$T$ is consistent."