Predicativity and Structuralism in Dedekind’s Construction of the Reals
It is a commonly held view that Dedekind’s construction of the real numbers is impredicative. This naturally raises the question of whether this impredicativity is justified by some kind of Platonism about sets. But when we look more closely at Dedekind’s philosophical views, his ontology does not look Platonist at all. So how is his construction justified? There are two aspects of the solution: one is to look more closely at his methodological views, and in particular, the places in which predicativity restrictions ought to be applied; another is to take seriously his remarks about the reals as things created by the cuts, instead of considering them to be the cuts themselves. This can lead us to make finer-grained distinctions about the extent to which impredicative definitions are problematic, since we find that Dedekind’s use of impredicative definitions in analysis can be justified by his (non-Platonist) philosophical views.
KeywordsRational Number Mathematical Object Irrational Number Philosophical View Initial Definition
I would like to thank Jeremy Avigad and Solomon Feferman for extremely helpful discussions about this paper, as well as an audience at the Society for Exact Philosophy meeting in Vancouver at which a preliminary version of this paper was presented. Finally, I would like to thank two anonymous referees for their constructive and insightful feedback on this paper.
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