### 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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