TY - JOUR
T1 - The number of normal measures
AU - Friedman, S. Y.David
AU - Magidor, Menachem
PY - 2009/9
Y1 - 2009/9
N2 - There have been numerous results showing that a measurable cardinal k can carry exactly α normal measures in a model of GCH, where a is α cardinal at most k++. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = k++, the maximum possible) and [1J (for α = k+, after collapsing k ++). In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of Mitchell order α), [2] (as in [12], but where k is the least measurable cardinal and α is less than k, starting with a measurable of high Mitchell order) and [11] (as in [12], but where k is the least measurable cardinal, starting with an assumption weaker than a measurable cardinal of Mitchell order 2). In this article we treat all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinaUty-preserving forcing. The proof uses k-Sacks forcing and the "tuning fork" technique of [8]. In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails.
AB - There have been numerous results showing that a measurable cardinal k can carry exactly α normal measures in a model of GCH, where a is α cardinal at most k++. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = k++, the maximum possible) and [1J (for α = k+, after collapsing k ++). In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of Mitchell order α), [2] (as in [12], but where k is the least measurable cardinal and α is less than k, starting with a measurable of high Mitchell order) and [11] (as in [12], but where k is the least measurable cardinal, starting with an assumption weaker than a measurable cardinal of Mitchell order 2). In this article we treat all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinaUty-preserving forcing. The proof uses k-Sacks forcing and the "tuning fork" technique of [8]. In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails.
UR - http://www.scopus.com/inward/record.url?scp=70349466237&partnerID=8YFLogxK
U2 - 10.2178/jsl/1245158100
DO - 10.2178/jsl/1245158100
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AN - SCOPUS:70349466237
SN - 0022-4812
VL - 74
SP - 1069
EP - 1080
JO - Journal of Symbolic Logic
JF - Journal of Symbolic Logic
IS - 3
ER -