Browsing by Author "Blanck, Rasmus"

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Contributions to the Metamathematics of Arithmetic: Fixed Points, Independence, and Flexibility
(2017-05-11) Blanck, Rasmus
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.
Metamathematical fixed points
(2011-05) Blanck, Rasmus
This thesis concerns the concept of metamathematical fixed points. After an introduction, we survey the field of metamathematics, from la fin du siècle to present. We are especially interested in the notion of fixed points, theorems on the existence of various kinds of fixed points, and their applications to metamathematics. The second part of the thesis is a technical investigation of sets of fixed points. Given some recursively enumerable, consistent extension T of Peano arithmetic, we define for each formula θ(x) the set Fix^T (θ) := {δ : T |- δ ↔ θ(δ)}. Our main result on these sets is that they are all Σ_1-complete. Furthermore, we define for each formula θ(x), the set Fix_Γ^T (θ) := {δ : T |- δ ↔ θ(δ)}, where δ is a sentence in Γ. Using methods of Bennet, Bernardi, Guaspari, Lindström, and Smullyan, we characterise these sets for formulas in Γ' ⊃ Γ, and provide partial results for formulas in Γ. We give a sufficient condition on recursive sets to be a set of fixed points, and show that such sets exists. We also present a sufficient condition for a recursively enumerable set of Γ-sentences to be a set of fixed points of a Γ-formula. In the following section, we study the structure of sets of fixed points ordered under set inclusion, and prove certain properties on these structures. Finally, we connect our research to another open problem of metamathematics, and state some possible further work.
Morley’s number of countable models
(2011-05-10) Blanck, Rasmus; Göteborgs universitet/Institutionen för lingvistik; Göteborg University/Department of Linguistics
A theory formulated in a countable predicate calculus can have at most 2א0 nonisomorphic countable models. In 1961 R. L. Vaught [9] conjected that if such a theory has uncountably many countable models, then it has exactly 2א0 countable models. This would of course follow immediately if one assumed the continuum hypothesis to be true. Almost ten years later, M. Morley [5] proved that if a countable theory has strictly more than א1 countable models, then it has 2א0 countable models. This leaves us with the possibility that a theory has exactly א1 , but not 2א0 countable models — and even today, Vaught’s question remains unanswered. This paper is an attempt to shed a little light on Morley’s proof.
On Rosser sentences and proof predicates
(2011-05-10) Blanck, Rasmus; Göteborgs universitet/Institutionen för filosofi, lingvistik och vetenskapsteori; Göteborg University/Department of Philosophy, Linguistics and Theory of Science
It is a well known fact that the G ̈del sentences γ of a theory T are o all provably equivalent to the consistency statement of T , Con T . This result is independent from choice of proof predicate. It has been proved by Guaspari and Solovay [4] that this is not the case for Rosser sentences of T . There are proof predicates whose Rosser sentences are all provably equivalent and also proof predicates whose Rosser sentences are not all provably equivalent. This paper is an attempt to investigate the matter and explicitly define proof predicates of both kinds.