Choiceless cardinals and the continuum problem

2022/01/27 by Gabriel Goldberg, Goldberg, Gabriel
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2201.11557

openalex publication_date 2022/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses so strong as to contradict the Axiom of Choice -- we solve several variants of the generalized continuum problem and identify structural features of the levels V_α of the cumulative hierarchy of sets that are eventually periodic, alternating according to the parity of the ordinal α. For example, if there is an elementary embedding from the universe of sets to itself, then for sufficiently large ordinals α, the supremum of the lengths of all wellfounded relations on V_α is a strong limit cardinal if and only if α is odd.

Related

• Even ordinals and the Kunen inconsistency
• Strong dichotomy of cardinality
• The Landscape of Large Cardinals
• Relations between some cardinals in the absence of the Axiom of Choice
• Disproof of the Continuum Hypothesis and Determination of the Cardinality of Continuum by Approximations of Sets