TY - JOUR
T1 - Possible size of an ultrapower of ω
AU - Jin, Renling
AU - Shelah, Saharon
PY - 1999/1
Y1 - 1999/1
N2 - Let ω be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In §1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of ω, whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In §2 we construct several λ-Archimedean ultrapowers of ω under some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exist a λ-Archimedean ultrapower of ω for some uncountable cardinal λ. This answers a question in [8], modulo the assumption of measurability.
AB - Let ω be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In §1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of ω, whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In §2 we construct several λ-Archimedean ultrapowers of ω under some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exist a λ-Archimedean ultrapower of ω for some uncountable cardinal λ. This answers a question in [8], modulo the assumption of measurability.
UR - http://www.scopus.com/inward/record.url?scp=0033479995&partnerID=8YFLogxK
U2 - 10.1007/s001530050115
DO - 10.1007/s001530050115
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AN - SCOPUS:0033479995
SN - 0933-5846
VL - 38
SP - 61
EP - 77
JO - Archive for Mathematical Logic
JF - Archive for Mathematical Logic
IS - 1
ER -