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The Cardinality of the Continuum is False in Non-Euclidean Geometries

Cardinality of the Continuum Continuum Hypothesis Parallel Postulate Fifth Postulate Euclidean Geometry Non-Euclidean Geometries Hyperbolic Geometry Elliptical Geometry Continuum Set.

Cite as:

Ayal Sharon (2020). The Cardinality of the Continuum is False in Non-Euclidean Geometries. RESEARCHERS.ONE, https://www.researchers.one/article/2020-04-7.


According to a proof in Euclidean geometry of the "Cardinality of the Continuum", that is attributed to Georg Cantor, a line has as many points as any line segment (not inclusive of the two end points). However, this proof uses parallel lines, and therefore assumes Euclid's Parallel Postulate as an axiom. But Non-Euclidean geometries have alternative axioms. 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. In Absolute geometry, neither Euclid's parallel postulate nor its alternatives are axioms. We provide an example in Spherical geometry and an example in Hyperbolic geometry wherein the "Cardinality of the Continuum" is false. Therefore the "Cardinality of the Continuum" is also false in Absolute geometry. So the "Continuum Hypothesis" is false too, because it assumes that the "Cardinality of the Continuum" is true.