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A Model-Theoretic Proof of Gödel's Theorem: Kripke's Notion of Fulfilment
Stockholm University, Faculty of Science, Department of Mathematics.
2017 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
Abstract [en]

The notion of fulfilment of a formula by a sequence of numbers, an approximation of truth due to Kripke, is presented and subsequently formalised in the weak arithmetic theory IΣ1, in some detail. After a number of technical results connecting the formalised notion to the meta-theoretical one a version of Gödel’s Incompleteness Theorem, that no consistent, recursively axiomatisable, Σ2-sound extension T of Peano arithmetic is complete, is shown by construction of a true Π2-sentence and a model of T where it is false, yielding its independence from T. These results are then generalised to a more general notion of fulfilment, proving that IΣ1 has no complete, consistent, recursively axiomatisable, Σ2-sound extensions by a similar construction of an independent sentence. This generalisation comes at the cost of some naturality, however, and an explicit falsifying model will only be obtained under additional assumptions.

The aim of the thesis is to reproduce in some detail the notions and results developed by Kripke and Quinsey and presented by Quinsey and Putnam. In particular no novel results are obtained.

Place, publisher, year, edition, pages
2017. , p. 90
Series
Självständiga arbeten i matematik ; 2017:3
Keywords [en]
Fulfilment, Gödel's Theorem, Incompleteness
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:su:diva-157490OAI: oai:DiVA.org:su-157490DiVA, id: diva2:1221351
Presentation
2017-01-13, Sal 32, Hus 5, Kräftriket, Stockholm, 13:00 (English)
Supervisors
Examiners
Available from: 2018-06-20 Created: 2018-06-19 Last updated: 2018-06-20Bibliographically approved

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