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HERBRAND’S THEOREM AND NON-EUCLIDEAN GEOMETRY

Published online by Cambridge University Press:  04 June 2015


MICHAEL BEESON
Affiliation:
SAN JOSÉ STATE UNIVERSITY SAN JOSE, CA USAE-mail: beesonpublic@gmail.com
PIERRE BOUTRY
Affiliation:
ICUBE, UNIVERSITY OF STRASBOURG CNRS, STRASBOURG FRANCEE-mail: boutry@unistra.fr
JULIEN NARBOUX
Affiliation:
ICUBE, UNIVERSITY OF STRASBOURG CNRS, STRASBOURG FRANCEE-mail: narboux@unistra.fr
Corresponding

Abstract

We use Herbrand’s theorem to give a new proof that Euclid’s parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof.


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Articles
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

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