Informatics, TU Vienna

Can Modalities Save Naive Set Theory?

In October of 2009 at Stanford University, the late Grigori “Grisha” Mints asked the senior author whether a naive set theory could be consistent in modal logic.

Abstract

In October of 2009 at Stanford University, the late Grigori “Grisha” Mints asked the senior author 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 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 may not be especially interesting. At that time Scott could not answer the consistency question, and neither could Mints, though they both agreed that a set theory based on that kind of comprehension would probably be very weak. And there, to the best of our knowledge, the problem sat ever since. Last November Scott received a notice from Carnegie Mellon that there would be a philosophy seminar on a naive set theory by 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 inconsistency for a strengthened version of comprehension, which did not quite work out, but the exchange became the basis for a paper. In the first draft of the paper, Scott and Lederman left open the consistency of the weaker comprehension, although they observed that it was not inconsistent by the analogue of the Russell set alone. In March of 2015 Liu approached them with a related model, which after a small correction gave a consistency proof. A few days later, Fritz approached them with essentially the same model.

Authors:  Peter Fritz, Harvey Lederman, Tiankai Liu, and Dana Scott

Note

This talk is organized by the Vienna Center for Logic and Algorithms and the Kurt Gödel Society.