Sh:835

• Shelah, S. PCF without choice. Arch. Math. Logic. To appear. arXiv: math/0510229
• 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 = cf(\mu) > \aleph_{0}, then from a well ordering of P(P(\kappa)) \cup {}^{\kappa >} \mu we can define a well ordering of {}^{\kappa} \mu.

• Version 2023-11-20 (32p)

@article{Sh:835,
 author = {Shelah, Saharon},
 title = {{PCF without choice}},
 journal = {Arch. Math. Logic},
 year = {to appear},
 note = {\href{https://arxiv.org/abs/math/0510229}{arXiv: math/0510229}},
 arxiv_number = {math/0510229}
}