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According Georg Cantor's "Cardinality of the Continuum", as proved in Euclidean geometry, any line segment (not inclusive of its two end points) has as many points as an entire line. This theorem is an underlying assumption of the "Continuum 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 "Continuum Hypothesis", that falsely assumes that the "Cardinality of the Continuum" is true. Concepts in number theory, such as the "number line" of real numbers, cannot be limited to Euclidean geometry. |
