The ‘Cardinality of the Continuum’ Is False in Non-Euclidean Geometries

Abstract

According Georg Cantor's Cardinality of the Continuum, as proved in Euclidean geometry, any segment (not inclusive of its two end points) has as many points as an entire line. This theorem is an underlying assumption of the Hypothesis, that conjectures that every infinite subset of the continuum of real numbers $\mathbb{R}$ is either equivalent to the set of natural numbers $\mathbb{N}$, or to $\mathbb{R}$ itself. However, the proof in Euclidean geometry uses parallel lines, and therefore assumes that Euclid's Parallel Postulate is true. But in Non-Euclidean geometries, Euclid's Parallel Postulate is false. In Hyperbolic geometry, at any point off of a given line, there are a plurality of lines parallel to the given line. In Elliptic geometry (which includes Spherical geometry), no lines are parallel, so two lines always intersect. Absolute geometry has neither Euclid's parallel postulate nor either of its alternatives. We provide examples in Spherical geometry and in Hyperbolic geometry where the Cardinality of the Continuum is false. Therefore it is also false in Absolute geometry. So it is a false proposition, as is the Hypothesis, that falsely assumes that the Cardinality of the Continuum is true. Concepts in theory, such as the number line of real numbers, cannot be limited to Euclidean geometry.