Provability logic

Stanford Encyclopedia of Philosophy (2008)

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Abstract
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.
Keywords provability logic  modal logic  self-reference  Peano Arithmetic  interpretability logic
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Transfinite Induction Within Peano Arithmetic.Richard Sommer - 1995 - Annals of Pure and Applied Logic 76 (3):231-289.

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