Easton functions and supercompactness

Brent Cody ; Sy-David Friedman ; Radek Honzik

Fundamenta Mathematicae, Tome 227 (2014), p. 279-296 / Harvested from The Polish Digital Mathematics Library

Résumé

Suppose that 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 < cf(F(α)), and (2) α < β ⇒ F(α) ≤ F(β). 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 $2^{δ} = F(δ)$ for every regular cardinal δ and κ remains λ-supercompact. This answers a question of [CM14].

Détail

Publié le : 2014-01-01

Zbl 06314013

EUDML-ID : urn:eudml:doc:282993