Fri Sep 25, 2015
60 Evans Hall, 4–6 PM
Dana Scott (UC Berkeley)
Can Modalities Save Naive Set Theory?
The late “Grisha” Mints once asked the speaker whether a naive set theory could be consistent in modal logic. Specifically he asked whether restricting the comprehension scheme to necessary properties was safe. Scott was working on a set theory in the Lewis system S4 of modal logic and Mints was happy to position his question in the same modal system. Obviously a very, very weak modality can avoid paradoxes, but such results are not especially interesting. At that time (2009) Scott could not answer the consistency question, and neither could Mints. Last November Scott noted that CMU Philosophy was hosting a seminar on a naive set theory by Harvey Lederman. Scott wrote him for his paper and said, “By the way, there is this question of Grisha Mints, and I wonder if you have an opinion?” Lederman sent back a sketch of a proof of the inconsistency of a strengthened version of comprehension. That proof at first did not quite work out, but was repaired in correspondence. Lederman mentioned the questions to two of his colleagues, and in March of 2015 Tiankai Liu suggested a possible model of a weaker comprehension scheme, which after a small correction gave a consistency proof. A few days later, Peter Fritz came up with essentially the same model. A paper has now been submitted for publication jointly by Fritz, Lederman, Liu, and Scott.