Metamathematical fixed points

Abstract

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.