One of the things Peacocke wants to do in “his book”:http://www.amazon.com/exec/obidos/ASIN/0521892430/caoineorg-20?creative=125581&camp=2321&link_code=as1 is distinguish his _moderate_ rationalism from Gödel’s _extreme_ rationalism.

Gödel thought we had a special faculty of insight into a priori domains, and that was how substantive a priori knowledge was possible. Peacocke, to his credit, doesn’t have any time for these mysterious faculties. But also, to his credit, he wants to directly address the arguments Gödel actually provides. In particular, he wants to address the argument Gödel raises against Carnap that careful reflection on, say, mathematics can reveal a priori truths that are not logical consequences of our previous discoveries/stipulations.

Gödel, if I remember correctly, was particularly interested in two cases of this. The first had to do with the “Gödel sentences” used in the incompleteness proof. The second had to do with set theory, and in particular axioms we might use to settle one way or the other the continuum hypothesis.

Peacocke thinks the kind of phenomena Gödel discusses, careful reflection leading to a priori discoveries, is real. But he thinks an account of a priori truth in terms of what it takes to possess certain concepts can account for it. And he goes through several instances of the phenomena showing how to do just this. But he doesn’t get to the two cases I mentioned above, which seem both the strongest for Gödel and the cases which he was most interested in.

To be fair he does discuss something like the incompleteness result in discussing the (allegedly a priori knowable) proposition _Every number has finitely many predecessors_. So maybe we can see how he’d handle the incompleteness cases. (Maybe.) But there’s no discussion of set theory at all, and I really have no idea how his approach would generalise to discussions of the continuum hypothesis. And this seems an unfortunate oversight in a discussion of Gödel.