On the number of cofinalities of cuts in ultraproducts of linear orders
Mohammad Golshani
TL;DR
This paper investigates the set of possible cofinalities of cuts in ultraproducts of a nondecreasing sequence of regular cardinals $\bar{a}=\langle \mu_i:i<\kappa\rangle$ indexed by a regular cardinal $\kappa$, denoted $Pcut(\bar{a})$. The main result is a sharp bound $|{\rm Pcut}(\bar{a})| \le 2^{\kappa}$, with a two-part argument showing $\lambda_2 \le 2^{\kappa}$ and $\lambda_1 \le 2^{\kappa}$ for any $(\lambda_1, \lambda_2) \in {\rm Pcut}(\bar{a})$, using an Erd\H{o}s--Rado partition argument for the second coordinate and an equivalence-class counting argument for the first. The paper further analyzes optimality: if $2^{\kappa}$ is not a fixed point of the aleph-function, the bound is strict; and it highlights that permitting $\lambda_2=1$ can realize larger sets, with forcing arguments showing $|{\rm Pcut}(\bar{a})|$ can be made as large as a prescribed inaccessible cardinal $\theta$ in suitable extensions. Overall, the work extends pcf-type analysis to cuts in ultraproducts, clarifying how cofinalities of cuts are constrained by $\kappa$ and by forcing, and illustrating the tightness of the $2^{\kappa}$ bound in various scenarios.
Abstract
Suppose $κ$ is a regular cardinal and $\bar a=\langle μ_i: i<κ\rangle$ is a non-decreasing sequence of regular cardinals. We study the set of possible cofinalities of cuts Pcut$(\bar a)=\{(λ_1, λ_2):$ for some ultrafilter $D$ on $κ$, $(λ_1, λ_2)$ is the cofinality of a cut of $\prod\limits_{i<κ} μ_i / D \}$.