Cofinality and measurability of the first three uncountable cardinals

Abstract

This paper discusses models of set theory without the Axiom of Choice. We investigate all possible patterns of the cofinality function and the distribution of measurability on the first three uncountable cardinals. The result relies heavily on a strengthening of an unpublished result of Kechris: we prove (under A D \mathsf {AD} ) that there is a cardinal κ \kappa such that the triple ( κ , κ + , κ + + ) (\kappa ,\kappa ^+,\kappa ^{++}) satisfies the strong polarized partition property.