Ordinal Functions

Abstract

In the Galvin-Hajnal theorem, the question of whether it holds for singular cardinals with countable cofinality was left open. Let ℵδ be singular and v be a cardinal. If δ = ℵδ, then \(X_\delta ^\nu = {\left| \delta \right|^\nu } < {X_{\left( {{{\left| \delta \right|}^v}} \right) + }}\) , and thus the Galvin-Hajnal theorem holds for the case that δ is a fixed point of the aleph function. Let us now turn to the case that δ < ℵδ. Then | δ | < ℵδ, and for every regular cardinal λ with | δ | < λ < ℵδ, the set a := [λ, ℵδ)reg is an interval of regular cardinals satisfying |a| < min(a), since |a| ≤ |δ|. Shelah defines an operator pcf which assigns to each set a of regular cardinals and each cardinal μ a set pcf μ (a) of regular cardinals satisfying the following properties for μ ≥ 1: a) a ⊆ pcfμ (a). b) min(a) = min pcfμ (a). c) If |a| < min(a), then |pcfμ (a)| ≤ |a|μ. d) If a is an interval of regular cardinals with |a| < min(a), then pcfµ(a) is an interval of regular cardinals. KeywordsMaximal IdealLinear ExtensionNormal SequenceSimple PropertyMain LemmaThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.