Set Theory New
Description
This paper introduces “Set Theory New: a revolutionary axiomatic framework designed to resolve the foundational crises inherent in stan dard Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC)[cite: 1]. By establishing eight indispensable axioms, including the Full Number Line Axiom (Axiom 1), this theory constructs an absolute, continuous, and gapless coordinate system for all real numbers and infinities[cite: 1]. A fundamental mathematical convention dictates that cardinality is strictly conserved based on unique foundational ele ments, leading directly to Theorem 17, which explicitly disproves Can tor’s Power Set Theorem by establishing Card(P(M)) = Card(M)[cite: 1]. Consequently, the combinatorial explosion of infinite subsets is proven mathematically false[cite: 1]. This absolute conservation in herently dismantles the existence of large cardinals, forcing exten sions, and pathological non-measurable sets[cite: 1]. Through the rigorous application of these new axioms, this paper systematically re solves and renders obsolete over 72 major contemporary open problems across abstract algebra, topology, model theory, and quantum physics, while mathematically disproving long-standing constructs such as the Continuum Hypothesis, Grothendieck Universes, and Mazur’s Torsion Theorem[cite: 1]