# Topološka potpunost logika dokazivosti

Mikec, Luka (2016) Topološka potpunost logika dokazivosti. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

System GLP is a system of modal logic introduced by Giorgi Japaridze in 1986. This system contains countably many modal operators $[n]$, one for each $n \in \omega$. Given a "sufficiently strong" theory of arithmetic A, system GLP provides a sound and complete description of a fragment of A. Most systems of modal logic can be given a certain type of semantics known as relational semantics (also known as Kripke semantics, or possible world semantics). In a certain sense, such semantics cannot be given to GLP in a satisfactory way. There are other types of semantics. In 2011, Lev Beklemishev and David Gabelaia proved that GLP is complete with respect to a certain class of polytopological spaces. The main goal of this thesis is to present the proof of topological completeness. In the first chapter, we work with relational semantics. This is mainly to showcase the issues that arise when trying to give GLP a (typical) relational interpretation. In the second chapter, we introduce topological constructions needed to prove the completeness theorem. These include lme-spaces, a certain well-behaved class of polytopological spaces. In the third chapter, we present the proof of completeness. This is done by exploiting relational semantics of another system of modal logic, known as system J. Theorems of GLP are, in a certain sense, reducible to theorems of J. It can be shown that for any J-sound relational frame there exists a lme-space and a validity-preserving morphism between them. By exploiting these facts, we can build a topological model that falsifies any formula that is not provable in GLP.