AI-Assisted Innovations on the Set of All Sets and the Continuum Hypothesis

This paper represents a human-AI collaboration to produce a self-referential logic and meta-mathematics.  Proofs and formal arguments are presented on the following within naïve set theory: (1) resolution of Russell’s Paradox; (2) resolution of Cantor’s Paradox; (3) resolution of the Burali-Forti Paradox; (4) creation of the set of all sets (SOAS) and a self-referential, self-ordinalizing set of all ordinals (SOAS); and (5) bijection of the SOAS and SOAO.  Additionally, within such a SOAS-SOAO, proofs and formal arguments are presented for the following: (6) bijection of ordinals and cardinals; (7) the well-ordering of the reals without the Axiom of Choice; (8) construction of the uncountable ordinals; (9) derivation of the natural numbers; (10) proof of Peano Arithmetic; (11) proof that the Continuum Hypothesis and Generalized Continuum Hypothesis are true; (12) proof of the consistency of Gödel’s Universe L and the SOAS-SOAO; (13) proof of a constructivist version of the Axiom of Choice; (14) resolution of the Banach-Tarski Paradox; (15) resolution of Vitali sets; (16) resolution of the Hausdorff Paradox; (17) resolution of Tarski’s circle-squaring problem; (18) resolution of non-measurable functions; (19) proof of the Jordan Curve Theorem; (20) proof of the Brouwer Fixed Point Theorem; and (21) proof of the Borsuk-Ulam Theorem.  There is also discussion about (a) the idea that V implies or can derive L, and (b) the SOAS-SOAO’s consistency with regards to large cardinals, transfinite cardinal arithmetic, and Grothendieck universes, given SOAS-SOAO’s consistency with Gödel’s L.  The findings above are all consistent with naïve set theory, classical logic, and each other.  The main enabling factor for the findings above is the resolution of self-referential logics via replacement of unary negation with cardinality logic at both the propositional logic level and in meta-logical syntactic architecture.  The work is part of a much larger series of AI-assisted findings in self-referential logic and mathematics.  The entire paper is LLM readable.