Intuitions of three kinds in Gödel's views on the continuum

Gödel’s views on mathematical intuition, especially as they are expressed in his well-known article on the continuum problem, have been much discussed, and yet some questions have perhaps not received all the attention they deserve. I will address two here. First, an exegetical question. Late in the paper Gödel mentions several consequences of the continuum hypothesis (CH), most of them asserting the existence of a subset of the straight line with the power of the continuum having some property implying the “extreme rareness” of the set. He judges all these consequences of CH to be implausible. The question I wish to consider is this: What is the epistemological status of Gödel’s judgments of implausibility supposed to be? In considering this question, several senses of “intuition” will need to be distinguished and examined. Second, a substantive question. Gödel makes much of the experience of the axioms of set theory “forcing themselves upon one as true,” and at least in the continuum problem paper makes this experience the main reason for positing such a faculty as “mathematical intuition.” After several senses of “intuition” have been distinguished and examined, however, I wish to address the question: In order to explain the Gödelian experience, do we really need to posit “mathematical intuition,” or will some more familiar and less problematic type of intuition suffice for the explanation? I will tentatively suggest that Gödel does have available grounds for excluding one more familiar kind of intuition as insufficient, but perhaps not for excluding another.