In this paper we study 1. the frame-theory of certain bimodal provability logics involving the reflection principle and we study2. certain specific bimodal logics with a provability predicate for a subtheory of Peano arithmetic axiomatized by a non-standardly finite number of axioms.

In this paper we study the reducibility order m (defined in a natural way) over n 0 -equivalence relations. In particular, for every n> 0 we exhibit n 0 -equivalence relations which are complete with respect to m and investigate some consequences of this fact (see Introduction).

We present some results on algebraic and modal analysis of polynomial distortions of the standard provability predicate in Peano Arithmetic PA, and investigate three provability-like modal systems related to the Gödel–Löb modal system GL. We also present a short review of relational and topological semantics for these systems, and describe the dual category of algebraic models of our main modal system.

Formulas of the propositional modal language with the unary modal operators □, Σ1, 1, Σ2, 2,… are considered as schemata of sentences of arithmetic , where □A is interpreted as “A is PA-provable”, ΣnA as “A is PA-equivalent to a Σn-sentence” and nA as “A is PA-equivalent to a Boolean combination of Σn-sentences”. We give an axiomatization and show decidability of the sets of the modal formulas which are schemata of: PA-provable, true arithmetical sentences.