Possible size of an ultrapower of \omega

Let 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 , 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 -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. Received: 19 November 1996     

Nice ℵ1 generated non-P-points,Part I

We define a family of non-principal ultrafilters on $\mathbb{N}$ which are, in a sense, very far from P-points. We prove the existence of such ultrafilters under reasonable conditions. In subsequent articles, we intend to prove that such ultrafilters may exist while no P-point exists. Though our primary motivations came from forcing and independence results, the family of ultrafilters introduced here should be interesting from combinatorial point of view too.