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Possible Size of an ultrapower of omega

Renling Jin·Saharon Shelah
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Abstract

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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Renling Jin, Saharon Shelah. 1998-01-15. Possible Size of an ultrapower of omega. https://arxiv.org/abs/math/9801153

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