Abstract

Let κ be an infinite cardinal. A κ-complete nonprincipal ultrafilter, or, for short, a κ- ultrafilter on a set A is a (nonempty) family U of subsets of A satisfying (i) S ⊆ U & |S|1 < κ ⇒ ∩S ∈ U (κ-completeness) (ii) X ∈ U & X ⊆ Y ⊆ A ⇒ Y ∈ U, (iii) ∀X ⊆ A [X ∈ U or A – X ∈ U] (iv) {a} ∉ U for any a..