It is our pleasure to announce that starting this October the Department of Philosophy, University of Warsaw will host a new mathematical logic seminar devoted to formal truth theories and implicit commitments of foundational theories as well as their conceptual surroundings. At least until the end of the year, the seminar will be held utterly virtually on a bi-weekly basis. We have the following permanent link to seminar meetings:
If you wish to be put on the mailing list, please contact Mateusz Łełyk (mlelyk(at)uw.edu.pl).
Time: Friday, April 9th, 8 pm (CEST, UTC+2) and Friday, April 23rd, 8 pm (CEST, UTC+2)
Speaker: Paul Gorbow
Title: The Copernican Multiverse of Sets
In these two talks, I will explain an untyped framework for the multiverse of set theory, developed in a joint paper with Graham Leigh. ZF is extended with semantically motivated axioms utilizing the new symbols Uni(U) and Mod(U, sigma), expressing that U is a universe and that sigma is true in the universe U, respectively. Here sigma ranges over the augmented language, leading to liar-style phenomena.
The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle, to the effect that the background theory of the multiverse does not hold a privileged position over the theories of its internal universes.
Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. This is applied to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory ZF, and thus not seriously limiting to the diversity of the set-theoretic multiverse. Considering truth-in-all-universes as an interpretation of the modal box-operator, it turns out that our main theory is consistent with the modal logics T and Gödel-Löb provability logic, but not with S4. I also plan to discuss connections with Hamkins’s multiverse theory, and the model of this constructed by Gitman and Hamkins.
I plan to reserve time throughout the talks for highlighting and inviting discussion about philosophical matters concerning our Copernican approach to the multiverse of sets.