We show the non-arithmeticity of 1st order theories of lattices of Σ n sentences modulo provable equivalence in a formal theory, of diagonalizable algebras of a wider class of arithmetic theories than has been previously known, and of the lattice of degrees of interpretability over PA. The first two results are applications of Nies’ theorem on the non-arithmeticity of the 1st order theory of the lattice of r.e. ideals on any effectively dense r.e. Boolean algebra. The theorem on degrees of (...) interpretability relies on an adaptation of techniques leading to Nies’ theorem. (shrink)

We prove a strengthened version of Shavrukov’s result on the non-isomorphism of diagonalizable algebras of two $\Sigma _1$ -sound theories, based on the improvements previously found by Adamsson. We then obtain several corollaries to the strengthened result by applying it to various pairs of theories and obtain new non-isomorphism examples. In particular, we show that there are no surjective homomorphisms from the algebra $(\mathfrak {L}_T, \Box _T\Box _T)$ onto the algebra $(\mathfrak {L}_T, \Box _T)$. The case of bimodal diagonalizable algebras (...) is also considered. We give several examples of pairs of theories with isomorphic diagonalizable algebras but non-isomorphic bimodal diagonalizable algebras. (shrink)

-/- Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. As a modal logic, provability logic has been studied since the early seventies, and has had important applications in the foundations of mathematics. -/- From a philosophical point of view, provability logic is interesting because (...) the concept of provability in a fixed theory of arithmetic has a unique and non-problematic meaning, other than concepts like necessity and knowledge studied in modal and epistemic logic. Furthermore, provability logic provides tools to study the notion of self-reference. (shrink)