Possible Size of an Ultrapower of ! 1

Let ! be the flrst inflnite ordinal (or the set of all natural numbers) with the usual order <. In x1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of !, 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 x2 we construct several ‚-Archimedean ultrapowers of ! under some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exist a ‚-Archimedean ultrapower of ! for some uncountable cardinal ‚. This answers a question in [8], modulo the assumption of measurability.