Abstract

The Continuum Hypothesis seems to be a counterexample to David Chalmers’s A Priori Scrutability thesis, according to which there is a compact class of truths from which all truths are a priori scrutable. Chalmers’s three-part answer to this problem runs as follows: either the Continuum Hypothesis is indeterminate; or adding a new axiom will settle the issue; or, if these two options do not work, we should add the Continuum Hypothesis to the scrutability base. I argue that Chalmers’s answer is unsatisfactory: the first horn of the trilemma can be interpreted in several ways, and either it departs from common mathematical practice and rests on weak analogies, or it shares the same problems with two other horns; the second horn does not provide good reasons to believe that from a fixed system of axioms all truths about our world are scrutable; the third horn of the trilemma renders Chalmers’s project empty.