Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata

1999/07/07 by Arthur W. Apter, Joel David Hamkins, Apter, Arthur W. +1
Computer Science · Mathematics · Psychology · #03E40 #03E55 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Philosophy and Theoretical Science #math.LO #msc:03E40 #msc:03E55

paper · pdf · doi:10.48550/arxiv.math/9907046

13 pages

arxiv created 1999/07/07 · openalex publication_date 1999/07/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.

Related

• Indestructibility and the level-by-level agreement between strong compactness and supercompactness
• The combinatorial essence of supercompactness
• The least weakly compact cardinal can be unfoldable, weakly measurable and nearly θ-supercompact
• Universal Indestructibility
• Exactly controlling the non-supercompact strongly compact cardinals