Axioms of elementary geometry and the Greek theorems
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.
scientific article; zbMATH DE number 994573