We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of $$\lambda $$ is well ordered for every $$\lambda $$ (really local version for a given $$\lambda $$ ). We think that in this frame much of pcf theory, (and combinatorial set theory in general) can be generalized. We prove here, in particular, that there is a proper class of regular cardinals, every large enough successor of singular is not measurable (...) and we can prove cardinal inequalities. Solving some open problems, we prove that if $$\mu> \kappa = \textrm{cf}(\mu ) > \aleph _{0},$$ then from a well ordering of $${\mathscr {P}}({\mathscr {P}}(\kappa )) \cup {}^{\kappa >} \mu $$ we can define a well ordering of $${}^{\kappa } \mu.$$. (shrink)

We examine the number of normal measures a successor cardinal can carry, in universes in which the Axiom of Choice is false. When considering successors of singular cardinals, we establish relative consistency results assuming instances of supercompactness, together with the Ultrapower Axiom (introduced by Goldberg in [12]). When considering successors of regular cardinals, we establish relative consistency results only assuming the existence of one measurable cardinal. This allows for equiconsistencies.