2013/11/01 by Brent Cody, Cody, Brent, Sy‐David Friedman +5
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.1311.0303
arxiv created 2013/11/01 · openalex publication_date 2013/11/01 · arxiv updated 2013/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose κ is λ-supercompact witnessed by an elementary embedding j:V→ M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton's theorem: (1) ∀α α<\textrmcf(F(α)) and (2) α<β \Longrightarrow F(α)≤ F(β). In this article we address the question: assuming GCH, what additional assumptions are necessary on j and F if one wants to be able to force the continuum function to agree with F globally, while preserving the λ-supercompactness of κ? We show that, assuming GCH, if F is any function as above, and in addition for some regular cardinal λ>κ there is an elementary embedding j:V→ M with critical point κ such that κ is closed under F, the model M is closed under λ-sequences, H(F(λ))⊆ M, and for each regular cardinal γ≤ λ one has (|j(F)(γ)|=F(γ))V, then there is a cardinal-preserving forcing extension in which 2δ=F(δ) for every regular cardinal δ and κ remains λ-supercompact. This answers a question of B. Cody, M. Magidor, On supercompactness and the continuum function, Ann. Pure Appl. Logic, (2013).