1949 B - CONTRADICTORITY OF ELEMENTARY GEOMETRY

Abstract

This chapter describes the contradictority of elementary geometry. The intersection theorem of Euclidean plane geometry which affirms that a common point can be found for any two lines a end l in the Euclidean plane that can neither coincide nor be parallel. As a consequence of the intersection theorem, it is found that for every line through P(0, 1), such that the tangent ρl of its angle with the axis of X satisfies −1 ≤ ρl < 0, a point S of intersection with the axis of X can be found, and therefore, a natural number n(l) such that xs < 2n(l) hence ρl < − 2−n(l), from which it follows that ρl < 0. The intersection theorem entails the equivalence of the relations which is contradictory, and consequently, the intersection theorem of plane Euclidean geometry is also contradictory. It may be expected that by means of species of checking numbers in connection with αf' resulting from the free variation of f, it will also be possible to prove the contradictority of other theories of classical mathematics, which have already been proved to be not true.