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Contributions to th...
Contributions to the Metamathematics of Arithmetic: Fixed Points, Independence, and Flexibility
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- Blanck, Rasmus, 1982 (författare)
- Gothenburg University,Göteborgs universitet,Institutionen för filosofi, lingvistik och vetenskapsteori,Department of Philosophy, Linguistics and Theory of Science
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(creator_code:org_t)
- ISBN 9789173469173
- Göteborg : Acta Universitatis Gothoburgensis, 2017
- Engelska.
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Serie: Acta philosophica Gothoburgensia/Acta Universitatis Gothoburgensis, 0283-2380
- Relaterad länk:
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https://gup.ub.gu.se...
Abstract
Ämnesord
Stäng
- This thesis concerns the incompleteness phenomenon of first-order arithmetic: no consistent, r.e. theory T can prove every true arithmetical sentence. The first incompleteness result is due to Gödel; classic generalisations are due to Rosser, Feferman, Mostowski, and Kripke. All these results can be proved using self-referential statements in the form of provable fixed points. Chapter 3 studies sets of fixed points; the main result is that disjoint such sets are creative. Hierarchical generalisations are considered, as well as the algebraic properties of a certain collection of bounded sets of fixed points. Chapter 4 is a systematic study of independent and flexible formulae, and variations thereof, with a focus on gauging the amount of induction needed to prove their existence. Hierarchical generalisations of classic results are given by adapting a method of Kripke’s. Chapter 5 deals with end-extensions of models of fragments of arithmetic, and their relation to flexible formulae. Chapter 6 gives Orey-Hájek-like characterisations of partial conservativity over different kinds of theories. Of particular note is a characterisation of partial conservativity over IΣ₁. Chapter 7 investigates the possibility to generalise the notion of flexibility in the spirit of Feferman’s theorem on the ‘interpretability of inconsistency’. Partial results are given by using Solovay functions to extend a recent theorem of Woodin.
Ämnesord
- HUMANIORA -- Filosofi, etik och religion -- Filosofi (hsv//swe)
- HUMANITIES -- Philosophy, Ethics and Religion -- Philosophy (hsv//eng)
- NATURVETENSKAP -- Matematik -- Algebra och logik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Algebra and Logic (hsv//eng)
Nyckelord
- arithmetic
- incompleteness
- flexibility
- independence
- non-standard models
- partial conservativity
- interpretability
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