Easton functions and supercompactness

Brent Cody, Sy-David Friedman, and Radek Honzik. "Easton functions and supercompactness." Fundamenta Mathematicae 226.3 (2014): 279-296. <http://eudml.org/doc/282993>.

@article{BrentCody2014,
abstract = {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].},
author = {Brent Cody, Sy-David Friedman, Radek Honzik},
journal = {Fundamenta Mathematicae},
keywords = {supercompactness; continuum function; easton's theorem},
language = {eng},
number = {3},
pages = {279-296},
title = {Easton functions and supercompactness},
url = {http://eudml.org/doc/282993},
volume = {226},
year = {2014},
}

TY - JOUR
AU - Brent Cody
AU - Sy-David Friedman
AU - Radek Honzik
TI - Easton functions and supercompactness
JO - Fundamenta Mathematicae
PY - 2014
VL - 226
IS - 3
SP - 279
EP - 296
AB - 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].
LA - eng
KW - supercompactness; continuum function; easton's theorem
UR - http://eudml.org/doc/282993
ER -