Local Properties

Abstract

In his book [Sh5], Shelah introduces the operator pcf_* which satisfies pcf_*(pcf_*(a)) = pcf_*(a). If b is a set of regular cardinals, which is not necessarily progressive, and if every limit point of b is a singular cardinal, then pcf_*(b) = pcf(b). Progressive sets a of regular cardinals have only singular limit points, so they satisfy pcf_*(a) = pcf(a). In this case, we even get pcf(a) = pcf_*(pcf(a)). If in addition every limit point of pcf(a) is singular, which is for example guaranteed by the condition |pcf(a)| < min(a), then pcf(pcf(a)) = pcf(a). A central tool for the proof of the hull property of pcf_* is the so-called localisation theorem: If c ⊑ pcf(a) is progressive and λ ∈ pcf(c), then there exists a set d such that d ⊑ c, |d| ≤ |a|, and λ ∈ pcf(d). This theorem will also be applied in the proof of a main result of pcf-theory: If a is a progressive interval of regular cardinals, then |pcf(a)| < |a|⁺⁴. The importance of this result will be demonstrated in Section 8.1.