2012/09/05 by Brent Cody, Cody, Brent
Computer Science · Mathematics · #03E35 #03E55 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #math.LO #msc:03E35 #msc:03E55
paper · pdf · doi:10.48550/arxiv.1209.1136
This is my dissertation, Advisor: Joel David Hamkins, 130 pages
arxiv created 2012/09/05 · openalex publication_date 2012/09/05 · arxiv updated 2012/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a Woodin cardinal δ, I show that if F is any Easton function with F"δ⊆δ and \GCH holds, then there is a cofinality-preserving forcing extension in which 2γ= F(γ) for each regular cardinal γ<δ, and in which δ remains Woodin. I also present a new example in which forcing a certain behavior of the continuum function on the regular cardinals, while preserving a given large cardinal, requires large cardinal strength beyond that of the original large cardinal under consideration. Specifically, I prove that the existence of a λ-supercompact cardinal κ such that \GCH fails at λ is equiconsistent with the existence of a cardinal κ that is λ-supercompact and λ++-tall. I generalize a theorem on measurable cardinals due to Levinski, which says that given a measurable cardinal, there is a forcing extension preserving the measurability of κ in which κ is the least regular cardinal at which \GCH holds. Indeed, I show that Levinski's result can be extended to many other large cardinal contexts. This work paves the way for many additional results, analogous to the results stated above for Woodin cardinals and partially supercompact cardinals.