Supercompactness and failures of GCH

Sy-David Friedman, and Radek Honzik. "Supercompactness and failures of GCH." Fundamenta Mathematicae 219.1 (2012): 15-36. <http://eudml.org/doc/286462>.

@article{Sy2012,
abstract = {Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver’s methods show that this can be done starting with κ being λ⁺⁺-supercompact (note that Silver’s result above is the special case when κ = λ). One can ask if there is an analogue of Woodin’s result for λ-supercompactness. We answer this question in the following strong sense: starting with the GCH and κ being λ-supercompact and λ⁺⁺-tall, we preserve λ-supercompactness of κ and kill the GCH at λ by directly manipulating the size of $2^\{λ\}$ (i.e. we do not force the failure of GCH at λ as a consequence of having $2^\{κ\}$ large enough). The direct manipulation of $2^\{λ\}$, where λ can be a successor cardinal, is the first step toward understanding which Easton functions can be realized as the continuum function on regular cardinals while preserving instances of λ-supercompactness.},
author = {Sy-David Friedman, Radek Honzik},
journal = {Fundamenta Mathematicae},
keywords = {lifting of embeddings; supercompact cardinals; Sacks forcing},
language = {eng},
number = {1},
pages = {15-36},
title = {Supercompactness and failures of GCH},
url = {http://eudml.org/doc/286462},
volume = {219},
year = {2012},
}

TY - JOUR
AU - Sy-David Friedman
AU - Radek Honzik
TI - Supercompactness and failures of GCH
JO - Fundamenta Mathematicae
PY - 2012
VL - 219
IS - 1
SP - 15
EP - 36
AB - Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver’s methods show that this can be done starting with κ being λ⁺⁺-supercompact (note that Silver’s result above is the special case when κ = λ). One can ask if there is an analogue of Woodin’s result for λ-supercompactness. We answer this question in the following strong sense: starting with the GCH and κ being λ-supercompact and λ⁺⁺-tall, we preserve λ-supercompactness of κ and kill the GCH at λ by directly manipulating the size of $2^{λ}$ (i.e. we do not force the failure of GCH at λ as a consequence of having $2^{κ}$ large enough). The direct manipulation of $2^{λ}$, where λ can be a successor cardinal, is the first step toward understanding which Easton functions can be realized as the continuum function on regular cardinals while preserving instances of λ-supercompactness.
LA - eng
KW - lifting of embeddings; supercompact cardinals; Sacks forcing
UR - http://eudml.org/doc/286462
ER -