Abstract

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 $${}^{\kappa } \mu.$$