David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Journal of Symbolic Logic 59 (3):737-756 (1994)
This paper discusses lower bounds for proof length, especially as measured by number of steps (inferences). We give the first publicly known proof of Gödel's claim that there is superrecursive (in fact. unbounded) proof speedup of (i + 1)st-order arithmetic over ith-order arithmetic, where arithmetic is formalized in Hilbert-style calculi with + and · as function symbols or with the language of PRA. The same results are established for any weakly schematic formalization of higher-order logic: this allows all tautologies as axioms and allows all generalizations of axioms as axioms. Our first proof of Gödel's claim is based on self-referential sentences: we give a second proof that avoids the use of self-reference based loosely on a method of Statman
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Citations of this work BETA
Cezary Cieśliński (2010). Deflationary Truth and Pathologies. Journal of Philosophical Logic 39 (3):325 - 337.
Jeffrey Ketland (2005). Some More Curious Inferences. Analysis 65 (285):18–24.
Matthias Baaz & Piotr Wojtylak (2008). Generalizing Proofs in Monadic Languages. Annals of Pure and Applied Logic 154 (2):71-138.
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