In this chapter I deal with results which involve mK and in which it cannot (so far as I know) be replaced by p. Once again I give first place to combinatorial results (§31). I follow this with two sections on measure theory and functional analysis (§§32–3); then I give part of S. Shelah's solution to Whitehead's problem (§34) and A.D. Taylor's and R. Laver's theorems on the regularity of ideals of Pω1 (§35).

Combinatorics

I begin with the fundamental theorem on extraction of sequences of directed sets from a partially ordered set satisfying Knaster's condition [31A], in a fairly general form, with its most commonly used corollaries [31B]. (Perhaps I should point out immediately that if m > ω1 then every ccc partially ordered set satisfies Knaster's condition [41A]; so that theorems which refer to mK and Knaster's condition can always be rewritten as theorems on ccc partially ordered sets, involving m). In 31C I describe a commonly-arising type of partially ordered set, the ‘S-respecting’ partial orders; I give lemmas for dealing with these [31Cb–31F], and an application [31G]. 31H is a powerful general theorem on the piecing together of almost-consistent functions. 31J is one of K. Kunen's theorems on ‘gaps’ in PN/[N]<ω. 31K concerns the cardinal mK itself.

Theorem

Let P be a partially ordered set satisfying Knaster's condition upwards.