Aksiomatizacije i modeli teorije NFU

doctoral thesis

• Title (english): Axiomatizations and models of theory NFU
• Author: Tin Adlešić
• Mentor: Tin Perkov (mentor)
• Mentor: Vedran Čačić (komentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Marko Horvat (član povjerenstva)
• Committee member: Marcel Maretić (član povjerenstva)
• Committee member: Miroslav Jerković (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-05-15, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Glavna tema disertacije je Quineova teorija skupova s atomima (NFU), koja se može smatrati vrstom teorije tipova. Disertacija je podijeljena na dva dijela. U prvom dijelu strogo formaliziramo sintaksu te razvijamo teoriju NFU. Zatim prikazujemo teoriju kardinalnih brojeva koristeći uređene parove Kuratowskog. Najvažniji doprinos ovog dijela je dokaz principa kardinalnog kvadriranja. Pomoću njega definiramo tipski ujednačene uređene parove te prezentiramo alternativnu aksiomatizaciju teorije NFU. Nakon toga, uvodimo osnove teorije ordinalnih brojeva u alternativnoj aksiomatizaciji i nastavljamo razvoj kardinalnih brojeva. Drugi dio prikazuje Boffinu konstrukciju modela teorije NFU pomoću modela Zermelo–Fraenkelove teorije skupova. Dokazujemo da Boffina konstrukcija zaista definira model teorije NFU, te zatim konstuiramo modele teorije NFU s aksiomom beskonačnosti te teorije NFU s aksiomom izbora. Na kraju pokazujemo kako konstruirati model teorije NFU s aksiomima beskonačnosti i izbora, a onda posljedično i alternativne aksiomatizacije.

Abstract (english)

The main topic of this dissertation is Quine’s set theory with atoms (NFU), which can be regarded as a form of type theory. The dissertation is divided into two parts. In the first part, we rigorously formalize the syntax and develop the theory NFU. Then we develop the theory of cardinal numbers, using Kuratowski’s ordered pairs. The main contribution of this part is a proof of the cardinal squaring principle. Using it, we define type-level ordered pairs and present the alternative axiomatic extension of NFU. After that, we introduce the basic concepts of ordinal numbers in the alternative axiomatization and continue to develop the theory of cardinal numbers. The second part is the exposition of Boffa’s construction of a model for NFU, using the model of Zermelo–Fraenkel’s set theory. We show that Boffa’s construction yields a model for NFU and we then construct a model for NFU with the axiom of infinity and NFU with the axiom of choice. In the end, we show how to construct a model for NFU with both the axiom of infinity and choice and then consequently for the alternative axiomatization.

• Keywords: teorija skupova; Quineova teorija; teorija modela
• Keywords (english): set theory; Quine’s theory; model theory
• Language: croatian
• URN:NBN: urn:nbn:hr:217:800738
• Project: Number: UIP-2017-05-9219; Title: Formalno rasuđivanje i semantike; Acronym: FORMALS; Leader: Tin Perkov; Jurisdiction: Croatia; Funder: Hrvatska zaklada za znanost; Funding stream: Installation Research Projects
• Study programme: Title: Doctoral study; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica u području prirodnih znanosti (doktor/doktorica u području prirodnih znanosti)
• Type of resource: Text
• Extent: vi, 141 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-05-21 09:51:09