Pcf without choice Sh835

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 \({}^{\kappa } \mu .\)