  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.
