Gentzen's remark has some bite in the case the standard first-order arithmetic PA, because we plausibly know a bit more arithmetically. But he apparently did not understand the generality of the incompleteness theorems: it holds for any theory which includes elementary arithmetic and happens to be consistent. In general, we just cannot see that they are consistent (even if they happen to be.) And as soon as Gentzen would define what exactly he means by "arithmetic means", we can prove that the consistency of the theory of those arithmetic means cannot be proved by those arithmetic means.

Gentzen's remark has some bite in the case the standard first-order arithmetic PA, because we plausibly know a bit more arithmetically. But he apparently did not understand the generality of the incompleteness theorems: they hold for any theory which includes elementary arithmetic and happens to be consistent. In general, we just cannot see that they are consistent (even if they happen to be.) And as soon as Gentzen would define what exactly he means by "arithmetic means", we can prove that the consistency of the theory of those arithmetic means cannot be proved by those arithmetic means.

Gentzen's remark has some bite in the case the standard first-order arithmetic PA. But he apparently does not understand the generality of the incompleteness theorems. As soon as Gentzen would define what exactly he means by "arithmetic means",

Gentzen's remark has some bite in the case the standard first-order arithmetic PA, because we plausibly know a bit more arithmetically. But he apparently did not understand the generality of the incompleteness theorems: it holds for any theory which includes elementary arithmetic and happens to be consistent. In general, we just cannot see that they are consistent (even if they happen to be.) And as soon as Gentzen would define what exactly he means by "arithmetic means", we can prove that the consistency of the theory of those arithmetic means cannot be proved by those arithmetic means.