Applications of pcf theory (Q2710599)

This paper contains several results in, and applications of, pcf theory, a powerful theory devised by the present author to obtain deep results in cardinal arithmetic [\textit{S. Shelah}, Cardinal arithmetic. Oxford University Press, Oxford (1994; Zbl 0848.03025)]. NEWLINENEWLINENEWLINEGiven a filter \(D\) on a cardinal \(\kappa\), and regular cardinals \(\lambda_i\) (\(i\in\kappa\)), say that \(P = \prod_{i\in\kappa} \lambda_i / D\) has true cofinality (tcf) \(\lambda\) if there is a sequence \(\langle f_\alpha ; \alpha < \lambda \rangle \subseteq \prod_{i\in\kappa} \lambda_i\) which is strictly increasing modulo \(D\) and cofinal in \(P\). Given an ordinal-valued function \(f\) on \(\kappa\), define \(T_D(f)\) as the minimal size of a family \(F \subseteq \prod_{i\in\kappa} f(i)\) of functions such that for every \(g \in \prod_{i\in\kappa} f(i)\) there is \(h \in F\) such that \(\{ h(i) \neq g(i) ; i \in \kappa \} \notin D\). The author proves that if \(D\) is \(\aleph_1\)-complete, \(\lambda > 2^\kappa\) is regular, and \(T_D (f) \geq \lambda\), then there are a set \(A \subseteq \kappa\) positive modulo \(D\) and regular cardinals \(\lambda_i\) in the interval \((2^\kappa , f(i)]\) (\(i\in A\)) such that \(\prod_{i \in A} \lambda_i / (D \upharpoonright A)\) has true cofinality \(\lambda\), thus establishing a connection between tcf and \(T_D\). NEWLINENEWLINENEWLINEAs an application, the author shows that if \(D\) and \(\lambda\) are as above and \(\mu_i > 2^\kappa\) (\(i \in \kappa\)) are cardinals, then the following are equivalent: (i) if \(B_i\) are Boolean algebras with Depth\(^+ (B_i) \geq \mu_i\), then Depth\(^+ (\prod_{i\in\kappa} B_i / D) > \lambda\); (ii) there are a set \(A \subseteq \kappa\) positive modulo \(D\) and regular cardinals \(\lambda_i < \mu_i\) such that tcf\((\prod_{i \in A} \lambda_i / (D \upharpoonright A)) = \lambda\). Here the depth (Depth\(^+ (B)\)) of the Boolean algebra \(B\) is the minimal \(\lambda\) such that there is no strictly increasing sequence of length \(\lambda\) in \(B\). NEWLINENEWLINENEWLINEFor a regular cardinal \(\lambda\) let \({\mathfrak{dp}}_\lambda\) be the minimal \(\mu\) such that there is no sequence \(\langle f_\alpha \in \lambda^\lambda ; \alpha < \mu \rangle\) which is strictly increasing modulo the cobounded filter. Also let \({\mathfrak a}_\lambda '\) be the minimal \(\mu\) such that there is no almost disjoint family of subsets of \(\lambda\) of size \(\mu\). It is shown that if \(D\) and \(\lambda\) are as before, \(\lambda_i > \kappa\) (\(i \in\kappa\)) are regular, \(\lambda = \) tcf\((\prod_{i \in \kappa} \lambda_i / D)\), \(\mu_i < {\mathfrak{dp}}_{\lambda_i}\) and \(\mu = \) tcf\((\prod_{i \in \kappa} \mu_i / D)\), then \(\mu < {\mathfrak{dp}}_{\lambda}\). A similar result is obtained for \({\mathfrak a}_\lambda '\) and for other cardinal invariants defined for any regular \(\lambda \). NEWLINENEWLINENEWLINEFurther results include a characterization of the supremum on the number of \(\kappa\)-branches of a tree of height \(\kappa\) with \(\lambda\) nodes in terms of pcf, an application of pcf to the existence of independent sets in stable theories, and a theorem saying that if \(D\) is an ultrafilter on \(\kappa\), \(\theta\) is minimal such that \(D\) is not \(\theta\)-regular and \(\mu = \mu^\theta \geq 2^\kappa\), then \(\mu\) can be represented as an ultraproduct of cardinals modulo \(D\).