Indestructibility of generically strong cardinals

Brent Cody, and Sean Cox. "Indestructibility of generically strong cardinals." Fundamenta Mathematicae 232.2 (2016): 131-149. <http://eudml.org/doc/286274>.

@article{BrentCody2016,
abstract = {Foreman (2013) proved a Duality Theorem which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of ω₁ is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie (2010)). As an application we prove that if ω₁ is generically strong, then it remains so after adding any number of Cohen subsets of ω₁; however many other ω₁-closed posets-such as Col(ω₁,ω₂)-can destroy the generic strongness of ω₁. This generalizes some results of Gitik-Shelah (1989) about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than ω₁.},
author = {Brent Cody, Sean Cox},
journal = {Fundamenta Mathematicae},
language = {eng},
number = {2},
pages = {131-149},
title = {Indestructibility of generically strong cardinals},
url = {http://eudml.org/doc/286274},
volume = {232},
year = {2016},
}

TY - JOUR
AU - Brent Cody
AU - Sean Cox
TI - Indestructibility of generically strong cardinals
JO - Fundamenta Mathematicae
PY - 2016
VL - 232
IS - 2
SP - 131
EP - 149
AB - Foreman (2013) proved a Duality Theorem which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of ω₁ is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie (2010)). As an application we prove that if ω₁ is generically strong, then it remains so after adding any number of Cohen subsets of ω₁; however many other ω₁-closed posets-such as Col(ω₁,ω₂)-can destroy the generic strongness of ω₁. This generalizes some results of Gitik-Shelah (1989) about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than ω₁.
LA - eng
UR - http://eudml.org/doc/286274
ER -