Generically extendible cardinals

2022/09/25 by Toshimichi Usuba, Usuba, Toshimichi
Computer Science · Mathematics · #03E40 #03E55 #03E57 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge

paper · pdf · doi:10.48550/arxiv.2209.12144

openalex publication_date 2022/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the notion of a generically extendible cardinal, which is a generic version of an extendible cardinal. We prove that the generic extendibility of ω₁ or ω₂ has small consistency strength, but that of a cardinal >ω₂ does not. We also consider some results concerned with generically extendible cardinals, such as indestructibility, generic absoluteness of the reals, and Boolean valued second order logic.

Related

• Weakly extendible cardinals and compactness of extended logics
• Extendible cardinals, and Laver-generic large cardinal axioms for extendibility
• Indestructibility of generically strong cardinals
• Usuba's extendible cardinal
• The first-order definability of generic large cardinals