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HERBRAND’S THEOREM AND NON-EUCLIDEAN GEOMETRY
Published online by Cambridge University Press: 04 June 2015
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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- Copyright © The Association for Symbolic Logic 2015
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