During the talk we discuss the lengths of proofs of arithmetical sentences in canonical truth theories. We focus mainly on extensions of the canonical compositional truth theory CT^-, introduced in the first talk of Bartosz Wcisło. In particular, we present a proof of the existence of a PTIME computable function f such that for every CT^- proof p of an arithmetical sentence A, f(p) is a PA-proof of A. Our proof relies on the arithmetization and uniformity of the Enayat-Visser model-theoretic construction of models with satisfaction classes. If time permits we shall comment on the speed-up phenomenon for various other truth theories, such as Kripke-Feferman (KF^-) and Friedman-Sheard (FS^-).
Logic seminar
usually takes place each Monday at 16:00 in IM, rear building, ground floorChair: Pavel Pudlak, Neil Thapen, Jan Krajíček
More information on the old seminar web page. The programme is announced via the mailing list.
Proof theoretic properties of truth predicates II
Mateusz Lelyk
Department of Philosophy, University of Warsaw
Monday, 17. May 2021 - 15:30 to 17:00
Proof theoretic properties of truth predicates I
Bartosz Wcislo
Institute of Mathematics, Polish Academy of Sciences
Monday, 10. May 2021 - 15:30 to 17:00
Axiomatic theory of truth is an area of logic which studies this notion in the following way: To a fixed base theory B (which in this talk will be Peano Arithmetic, PA), we add a fresh predicate T(x) with the intended reading "x is (a Goedel code of) a true sentence" and axioms which guarantee that the predicate T has a truth-like behaviour. For instance, we can stipulate that T satisfies Tarskian compositional conditions for arithmetical sentences (obtaining a theory called CT^-) or that it satisfies various forms of induction.
It is a classical result that the theory CT^- with the full induction scheme for the formulae containing the truth predicate, called CT, is not conservative over PA. In fact, it proves so called global reflection principle which says that an arbitrary arithmetical sentence provable in PA is true (in the sense of the truth predicate). On the other hand, a theorem by Kotlarski, Krajewski, and Lachlan shows that CT^- is conservative.
In the... more
Feasible disjunction property for intuitionistic modal logics
Raheleh Jalali
Utrecht University
Monday, 26. April 2021 - 15:30 to 17:00
In this talk, we present a uniform method to prove feasible disjunction property (DP) for various intuitionistic modal logics. More specifically, we prove that if the rules in a sequent calculus for a modal intuitionistic logic have a special form, then the sequent calculus enjoys feasible DP. Our method is essentially an adaptation of the method used by Hrubes in his lower bound proof for the intuitionistic Frege system. As a consequence, we uniformly prove that the sequent calculi for intuitionistic logic, the intuitionistic version of several modal logics such as K, T, K4, S4, S5, their Fisher-Servi versions, propositional lax logic, and many others have feasible DP. Our method also provides a way to prove negative results: we show that any intermediate modal logic without DP does not have a calculus of the given form. This talk is based on a joint work with Amir Tabatabi.
Compactness at small cardinals
Radek Honzik
Charles University
Monday, 12. April 2021 - 15:30 to 17:00
We will survey some results related to compactness principles at small cardinals which extend the usual first-order compactness to more complex structures.
More specifically, suppose kappa is an uncountable regular cardinal (typically kappa can be taken to be the size of the reals). We will review a variety of compactness principles, such as the tree property, stationary reflection, Rado's conjecture, etc., which claim that if all parts of size < kappa of a given structure of size kappa have some property, so does the whole structure.
We will discuss basic models in which such principles hold, consistency strength of these principles, implications between the principles and other hypotheses (such as CH), and some consequences.