Axioms of elementary geometry and the Greek theorems
type: Item
instance of: scholarly article
title: Axioms of elementary geometry and the Greek theorems
zbMATH Open document ID: 0872.51014
published in: Mathematische Semesterberichte
publication date: 1997-08-11T00:00:00Z
review text: This is a two-part paper. The first (considered by its author to be the raison d'être of the paper) is an epistemological discussion of the interplay between intuition and logic. The second is a non-elementary axiomatization of 2-dimensional Euclidean geometry. The axiomatics takes both set theory and the field of real numbers for granted. It also contains `descriptive' axioms, like ``\(E\) is a non-empty set. The elements of \(E\) are points.'' It first axiomatizes the real affine plane, and then the Euclidean plane by axioms that define for every real number \(\theta\) the rotation of angle \(\theta\) (in the positive trigonometric sense) about a fixed point \(O\). It is claimed that Gödel proved that the consistency of Euclidean, hyperbolic, and elliptic geometry cannot be proved. This is true only if it refers to the second-order version of them or to a first-order version which includes an axiom system for set theory. There is no consistency problem for the first-order complete axiomatizations of these geometries, which was carried out by Tarski. The consistency cannot be proved inside these first-order theories since the provability predicate is not definable therein, but there is no doubt whatsoever about their consistency, for they are, as shown by Tarski, decidable.
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 03A05
zbMATH DE Number: 994573
zbMATH Keywords: elementary geometry
zbMATH Keywords: 2-dimensional Euclidean geometry
zbMATH Keywords: axiomatics
author: Hans Grauert
reviewed by: Victor V. Pambuccian
MaRDI profile type: Publication
full work available at URL: s005910050022
OpenAlex ID: W2000463251
DOI: 10.1007/S005910050022
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description: scientific article; zbMATH DE number 994573