Cofinality and Measurability of the First Three Uncountable Cardinals
Arthur W. Apter, Stephen C. Jackson, Benedikt Löwe

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unpublished result of Kechris: we prove (under AD) that there is a
cardinal κ such that the triple (κ, κ+ , κ++) satisfies the strong
polarized partition property.

Keywords: Axiom of Choice, Cardinals, Partition Properties