Possible Size of an ultrapower of $\omega$

by Jin and Shelah. [JiSh:626]
Archive for Math Logic, 1999
Let omega be the first infinite ordinal (or the set of all natural numbers) with the usual order < . In section 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of omega, whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [CK], modulo the assumption of supercompactness. In section 2 we construct several lambda-Archimedean ultrapowers of omega under some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exist a lambda-Archimedean ultrapower of omega for some uncountable cardinal lambda . This answers a question in [KS], modulo the assumption of measurability.


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