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June 2007 In the shadows of the Löwenheim—Skolem theorem: Early combinatorial analyses of mathematical proofs
Jan von Plato
Bull. Symbolic Logic 13(2): 189-225 (June 2007). DOI: 10.2178/bsl/1185803805

Abstract

The Löwenheim—Skolem theorem was published in Skolem’s long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal notation is given here by which these proofs become transparent. A third section of Skolem’s paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem’s result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved.

The forgotten parts of Skolem’s first paper on the Löwenheim—Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results.

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Jan von Plato. "In the shadows of the Löwenheim—Skolem theorem: Early combinatorial analyses of mathematical proofs." Bull. Symbolic Logic 13 (2) 189 - 225, June 2007. https://doi.org/10.2178/bsl/1185803805

Information

Published: June 2007
First available in Project Euclid: 30 July 2007

zbMATH: 1139.03004
MathSciNet: MR2322079
Digital Object Identifier: 10.2178/bsl/1185803805

Rights: Copyright © 2007 Association for Symbolic Logic

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Vol.13 • No. 2 • June 2007
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