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

我们主要研究具有限制选择的集合论模型，例如，ZF + DC + \(\lambda \)的可数子集族对于每个\(\lambda \)都是良序的（给定\(\ ) 的真正本地版本拉姆达 \) )。我们认为在这个框架中，大部分 PCF 理论（以及一般的组合集合论）都可以推广。我们在这里特别证明，存在一类正规基数，每个足够大的单数后继都是不可测的，并且我们可以证明基数不等式。解决一些悬而未决的问题，我们证明如果\(\mu> \kappa = \textrm{cf}(\mu ) > \aleph _{0},\)那么从\({\mathscr {P} }({\mathscr {P}}(\kappa )) \cup {}^{\kappa >} \mu \)我们可以定义\({}^{\kappa } \mu .\)的良好排序