Provability logic

Provability logic is a branch of mathematical logic that studies formal systems of provability. Specifically, it deals with the properties and behaviors of provability predicates, which are statements or operators that express the idea that a certain statement is provable within a given formal system. One of the most prominent systems within provability logic is known as Gödel's provability logic, often represented by the modal system \( GL \) (Gödel-Löb logic).

The finite model property is a concept in mathematical logic, specifically in model theory, that refers to the characteristics of certain logical theories regarding their models. A theory (which is a set of sentences in a formal language) is said to have the finite model property if every finite model of the theory can be extended to an infinite model. For a more formal definition, consider a theory \( T \) in a first-order logic.

Interpretability logic is a subfield of logic that focuses on understanding and formalizing the concept of interpretability between different mathematical structures or theories. The core idea is to explore how one theory can be interpreted in terms of another, investigating the relationships between them and the information that can be derived from such interpretations. This area of study often involves the use of formal logic to specify how the elements and operations of one structure can be represented within another.