Corpus ID: 237491961

Provability Logic: models within models in Peano Arithmetic

  title={Provability Logic: models within models in Peano Arithmetic},
  author={Alessandro Berarducci and Marcello Mamino},
In 1994 Jech gave a model-theoretic proof of Gödel’s second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech’s proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without passing through the arithmetized completeness theorem… Expand


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This note presents a solution of the previous problem with respect to the system Z μ, and, more generally, to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function ( k, l ) used below is definable. Expand
On Gödel’s second incompleteness theorem
In this note we give a short proof of Godel's Second Incompleteness Theorem. G6del's Second Incompleteness Theorem states that no sufficiently strong consistent mathematical theory can prove its ownExpand
Provability interpretations of modal logic
We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and letExpand
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This chapter discusses Arithmetic as Number Theory, Set Theory and Logic, Fragments and Combinatorics, and Models of Fragments of Arithmetic. Expand
‘Fuzzy logic’ means different things to different people. For some it is a philosophy of life – “a way to break the stranglehold that the black-and-white thinking of the Western tradition has uponExpand
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This chapter is dedicated to the memory of George Boolos, the prime inspirer of the work in the logic of provability, from the start of the subject until his death on 27 May 1996. Expand
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New proofs of the incompleteness theorems, found in 1990s, do not require the diagonal lemma as a method of construction of an independent statement. Expand
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