Skupovne interpretacije modalne logike

master's thesis

• Author: Filip Kiršek
• Mentor: Mladen Vuković (mentor)
• Committee member: Mladen Vuković (predsjednik povjerenstva)
• Committee member: Vedran Čačić (član povjerenstva)
• Committee member: Tin Perkov (član povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Modalna logika se bavi idejama nužnosti i mogućnosti. U ovom radu definiramo modalne sustave K, K4, GL i I, te dokazujemo standardne teoreme adekvatnosti i potpunosti tih sustava sa Kripkeovom semantikom. Zatim ponovno predstavljamo aksiome Zermelo-Fraenkelove teorije skupova, te uvodimo pojam relativizacije formule, te pojam tranzitivnih modela. Potom predstavljamo pregled ideje Gödelove aritmetizacije sustava Peanovih aksioma i Zermelo-Fraenkelove teorije skupova. Koristeći te pojmove razmatramo kako teorija ZF može govoriti o dokazivosti te relativnoj nezavisnosti i konzistentnosti svojih aksioma. Konačno, iskazujemo Solovayeve teoreme aritmetičke potpunosti za modalne sustave GL, GLS, I i J, te predstavljamo skicu dokaza aritmetičke potpunosti za sustav I. Za upotpunjavanje te skice dokaza je potrebna, među ostalima, metoda forcinga.

Abstract (english)

Modal logic is concerned with the notions of necessity and possibility. This thesis defines the modal systems K, K4, GL and I, and contains proofs of soundness and completeness theorems of those systems in standard Kripkean semantics. The axioms of the Zermelo-Fraenkel set theory are then presented, and the notions of relativizations and transitive models of ZF are defined. The idea behind Gödel numbering of PA and ZF formulas is presented as well. Using those notions, the idea of formalizing the method of ZF proving relative consistency and independence of its axioms is described. Finally, Solovay's theorems of arithmetical completeness for modal systems GL, GLS, I and J are then presented, and a rough overview of the proof of arithmetical completeness of system I, concerning transitive models of ZF, is explored. The notion of forcing, among other ideas, is needed for completing the proof.

• Keywords: modalna logika; modalni sustavi; Kripkeova semantika; Zermelo-Fraenkelova teorija skupova; Gödelova aritmetizacija sustava Peanovih aksioma; Solovayevi teoremi aritmetičke potpunosti za modalne sustave
• Keywords (english): modal logic; modal systems; Kripkean semantics; Zermelo-Fraenkel set theory; Gödel numbering of PA and ZF formulas; Solovay's theorems of arithmetical completeness for modal systems
• Language: croatian
• URN:NBN: urn:nbn:hr:217:846037
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra računarstva i matematike (magistar/magistra računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-02-12 12:39:57