Pcf without choice Sh835

We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of λ is well ordered for every λ (really local version for a given λ ). 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 μ> κ = cf(μ ) > ℵ _0, then from a well ordering of 𝒫(𝒫(κ )) ∪^κ >μ we can define a well ordering of ^κμ .