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University of Gothenburg / Göteborgs Universitet

1. Blanck, Rasmus. Contributions to the Metamathematics of Arithmetic: Fixed Points, Independence, and Flexibility.

Degree: 2017, University of Gothenburg / Göteborgs Universitet

URL: http://hdl.handle.net/2077/52271

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.

Subjects/Keywords: arithmetic; incompleteness; flexibility; independence; non-standard models; partial conservativity; interpretability

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APA (6^{th} Edition):

Blanck, R. (2017). Contributions to the Metamathematics of Arithmetic: Fixed Points, Independence, and Flexibility. (Thesis). University of Gothenburg / Göteborgs Universitet. Retrieved from http://hdl.handle.net/2077/52271

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Blanck, Rasmus. “Contributions to the Metamathematics of Arithmetic: Fixed Points, Independence, and Flexibility.” 2017. Thesis, University of Gothenburg / Göteborgs Universitet. Accessed January 21, 2018. http://hdl.handle.net/2077/52271.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Blanck, Rasmus. “Contributions to the Metamathematics of Arithmetic: Fixed Points, Independence, and Flexibility.” 2017. Web. 21 Jan 2018.

Vancouver:

Blanck R. Contributions to the Metamathematics of Arithmetic: Fixed Points, Independence, and Flexibility. [Internet] [Thesis]. University of Gothenburg / Göteborgs Universitet; 2017. [cited 2018 Jan 21]. Available from: http://hdl.handle.net/2077/52271.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Blanck R. Contributions to the Metamathematics of Arithmetic: Fixed Points, Independence, and Flexibility. [Thesis]. University of Gothenburg / Göteborgs Universitet; 2017. Available from: http://hdl.handle.net/2077/52271

Not specified: Masters Thesis or Doctoral Dissertation