Quineovi ''Novi temelji'' teorije skupova

master's thesis

• Author: Gabrijela Zlatunić
• Mentor: Vedran Čačić (mentor)
• Committee member: Vedran Čačić (predsjednik povjerenstva)
• Committee member: Matej Mihelčić (član povjerenstva)
• Committee member: Goranka Nogo (član povjerenstva)
• Committee member: Marko Horvat (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-22, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Dajemo kratki pregled ovoga rada. Najprije uvodimo jezik teorije \(\mathsf{NF}\) i sve osnovne pojmove na kojima je ona zasnovana. Zatim definiramo osnovne operacije na skupovima i uvodimo skupove od značaja poput skupa prirodnih brojeva i skupa kardinalnih brojeva. Želimo, barem ukratko, predstaviti ljepotu i gracioznost teorije \(\mathsf{NF}\) zasnovane na jednostavnosti. Njena veličina ne bi bila upitna da nemamo drugo poglavlje - ono s druge strane preslikava sivu notu same teorije objašnjavajući, nadamo se dovoljno detaljno, zašto ista nije zaživjela, barem ne u svom prvotnom obliku. Težina Speckerovog teorema ne leži samo u dokazu (koliko god to bilo kontradiktorno veličini samog dokaza), nego i u posljedicama koje je tvrdnja imala na teoriju. Međutim, kako treće poglavlje da naslutiti, teorija je ipak bila dovoljno intrigantna i zanimljiva da je ovakav udarac nije slomio i bacio u zaborav. Tek nekolicina promjena u stvaranju jezika teorije rezultirala je (ne samo jednom) verzijom teorije koja ne samo da je relativno konzistentna s nekom već prihvaćenom teorijom u matematici, već ostaje konzistentnom i dodajući joj aksiom izbora i aksiom beskonačnosti - dvije tvrdnje nepomirljive u čistoj \(\mathsf{NF}\).

Abstract (english)

We provide a brief overview of this work. Firstly, we introduce the language of NF theory and all the fundamental concepts on which it is based. Then, we define basic operations on sets and introduce sets of significance such as the set of natural numbers and the set of cardinal numbers. We have attempted, at least briefly, to convey the beauty and elegance of NF theory based on its simplicity. Its magnitude would not be questionable if we did not have the second chapter, which, on the other hand, casts a shadow over the theory itself-explaining, hopefully in sufficient detail, why it has not caught on, at least not in its original form. The weight of Specker’s theorem lies not only in its proof (in spite of its size) but in the consequences its assertion has on the theory. However, as hinted at in the third chapter, the theory is intriguing and interesting enough that such a blow does not break it and send it to oblivion. Only a few changes in the formulation of the theory’s language result in multiple versions of the theory that are not only relatively consistent with previously accepted ones in mathematics, but also remain consistent when adding the axiom of choice and the axiom of infinity—two claims irreconcilable in pure NF.

• Keywords: jezik teorije; Speckerov teorem; aksiom beskonačnosti
• Keywords (english): language of NF theory; Specker’s theorem; axiom of infinity
• Language: croatian
• URN:NBN: urn:nbn:hr:217:133180
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar računarstva i matematike (sveučilišni magistar računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-07 12:30:59