The PCF Theorem Revisited

Summary

The pcf theorem (of the possible cofinability theory) was proved for reduced products $\prod\limits_{i < k} {{\lambda _i}/I}$, where κ < min _i<κ λ _i . Here we prove this theorem under weaker assumptions such as wsat(I) < min _i<κ λ _i , where wsat(I) is the minimal θ such that κ cannot be divided to θ sets ∉ I (or even slightly weaker condition). We also look at the existence of exact upper bounds relative to < _I (< _I —eub) as well as cardinalities of reduced products and the cardinals T _D (λ). Finally we apply this to the problem of the depth of ultraproducts (and reduced products) of Boolean algebras.

Partially supported by the Deutsche Forschungsgemeinschaft, grant Ko 490/7–1. Publication no. 506.