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Transactions of the American Mathematical Society

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Cycling in proofs and feasibility


Author: A. Carbone
Journal: Trans. Amer. Math. Soc. 352 (2000), 2049-2075
MSC (1991): Primary 03F07, 03F20, 03F05, 03H15, 68R10
DOI: https://doi.org/10.1090/S0002-9947-99-02300-4
Published electronically: September 8, 1999
MathSciNet review: 1603882
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Abstract: There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh's lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles.


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Additional Information

A. Carbone
Affiliation: Math\a’ematiques/Informatique, Universit\a’e de Paris XII, 61 Avenue du G\a’en\a’eral de Gaulle, 94010 Cr\a’eteil Cedex, France
Email: ale@gauss.math.jussieu.fr, ale@univ-paris12.fr

DOI: https://doi.org/10.1090/S0002-9947-99-02300-4
Keywords: Feasible numbers, cut elimination, cycles in proofs, structure of proofs, complexity of proofs.
Received by editor(s): June 13, 1997
Received by editor(s) in revised form: January 21, 1998
Published electronically: September 8, 1999
Additional Notes: Partially supported by the Lise-Meitner Stipendium # M00187-MAT (Austrian FWF)
Article copyright: © Copyright 2000 American Mathematical Society