Ordinal Functions

Abstract

In the Galvin-Hajnal theorem, the question of whether it holds for singular cardinals with countable cofinality was left open. Let ℵ_δ be singular and v be a cardinal. If δ = ℵ_δ, 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 < ℵ_δ. Then | δ | < ℵ_δ, and for every regular cardinal λ with | δ | < λ < ℵ_δ, the set a := [λ, ℵ_δ)_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 ⊆ pcf_μ (a).

• b) min(a) = min pcf_μ (a).

• c) If |a| < min(a), then |pcf_μ (a)| ≤ |a|^μ.

• d) If a is an interval of regular cardinals with |a| < min(a), then pcf_µ(a) is an interval of regular cardinals.