Submitted on 2020-04-04

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.

Submitted on 2020-04-04

Quantum Electrodynamics (QED) Renormalizaion is a logical paradox, and thus is mathematically invalid. It converts divergent series into finite values by use of the Euler-Mascheroni constant. The definition of this constant is a conditionally convergent series. But the Riemann Series Theorem proves that any conditionally convergent series can be rearranged to be divergent. This contradiction (a series that is both convergent and divergent) violates the Law of Non-Contradiction (LNC) in "classical" and intuitionistic logics, and thus is a paradox in these logics. This result also violates the commutative and associative properties of addition, and the one-to-two mapping from domain to range violates the definition of a function in Zermelo-Fraenkel set theory.

In addition, Zeta Function Regularization is logically and mathematically invalid. It equates two definitions of the Zeta function: the Dirichlet series definition, and Riemann's definition. For domain values in the half-plane of "analytic continuation", the two definitions contradict: the former is divergent and the latter is convergent. Equating these contradictory definitions there creates a paradox (if both are true), or is logically invalid (if one is true and the other false). We show that Riemann's definition is false, because its derivation includes a contradiction: the use of both the Hankel contour and Cauchy's integral theorem. Also, a third definition of the Zeta function is proven to be false. The Zeta function is exclusively defined by the Dirichlet series, which has no zeros (and therefore the Riemann hypothesis is a paradox).

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