Against the viability of the a priori

1. If the notion of the a priori is viable, there is a recursive logical system S, whose soundness is a priori, such that all a priori mathematical truths are provable in S.
2. If the notion of the a priori is viable, then the truth of Peano Arithmetic (PA) is a priori.
3. If the notion of the a priori is viable, then it is a priori that whatever is true is consistent in every sound logical system.

Assume the notion of the a priori is viable. By (1) and (2), PA is provable in S. By (3), the consistency in S of PA is a priori, and hence the consistency of S of PA is provable in S by (1). These conclusions contradict the soundness of S by Goedel's Second Incompeteness Theorem.

There are two ways of taking the above argument. One can take it as concerning the a priori as such, or the a priori for humans. Either way, the premises are plausible.

I think the main controversy is going to be about (1). To deny (1) would be to hold that there are infinitely many mathematical truths, not a priori reducible to a finite number of a priori assumptions, but that are nonetheless a priori. This is particularly weird if the a priori is a priori for us: Do we really have some mysterious inner capacity to cognize irreducibly infinitely many mathematical truths a priori? But even if the a priori is not relativized to humans, it's weird. In just what sense are all these mathematical claims a priori?

Let me sharpen the last point. We can restrict our attention to those mathematical truths that are formulable in our mathematical vocabulary, since these are the only ones that come up in the argument. But the a priori truths formulable in a given vocabulary seem to be basically the analytic ones. Are there really infinitely many independent analytic truths formulable in our mathematical vocabulary? Are we that rich that so much is implicit in what we say?

I think this does serious damage to the Chalmers project.