Possible size of an ultrapower of $\omega$

Abstract.

Let $\omega$ 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 $\omega$, 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 $\lambda$-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 [8], modulo the assumption of measurability.