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Essential incompleteness in propositional logic

Posted on May 21, 2020July 23, 2020 by Andrei Sipoș

Consider the following formulation of Gödel’s first incompleteness theorem. Theorem. Any complete, consistent theory that extends PA is not recursive. In other words, PA is essentially incomplete. Five years ago, I wondered whether such a recursively enumerable and essentially incomplete theory could be easily found in classical propositional logic, where formulas are built up from…

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Continuity in System T

Posted on May 15, 2020May 27, 2020 by Thomas Powell

Continuity is one of the most important concepts in higher-order computability. Informally, a functional is continuous if we only require a finite part of its input to compute a finite part of its output. Though there are a number of ways of making this precise, at type level 2, that is, for functionals , continuity…

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‘Simple’ proofs of cut-elimination: classical logic

Posted on May 7, 2020October 20, 2022 by Anupam Das

In the last couple years, I have taught cut-elimination several times to research-level students (ESSLLI ’18, LSS ’18, and to PhD students at the University of Birmingham). Every time I am left squirming when I have to deal with the monster in the closet: . Most cut-elimination proofs do something like this: Main induction on…

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Welcome to The Proof Theory Blog!

Posted on May 1, 2020May 27, 2020 by Anupam Das

Welcome to The Proof Theory Blog! The purpose of this website is to give proof theorists a venue to communicate ideas, works-in-progress, gems, or simply observations that may be relevant to the proof theory community. The hope is that it can eventually evolve into a vibrant forum for proof theoretic discussions and collaboration. Proof theory…

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