We consider mainly the following version of set theory: “ZF+DC and for every λ,λℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda, \lambda^{\aleph_0}}$$\end{document} is well ordered”, our thesis is that this is a reasonable set theory, e.g. on the one hand it is much weaker than full choice, and on the other hand much can be said or at least this is what the present work tries to indicate. In particular, we prove that for a sequence δ¯=⟨δs:s∈Y⟩,cf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\delta} = \langle\delta_{s}: s \in Y\rangle, {\rm cf}}$$\end{document} large enough compared to Y, we can prove the pcf theorem with minor changes.We then deduce the existence of covering numbers and define and prove existence of a class of true successor cardinals. Using this we give some diagonalization arguments on Abelian groups, chosen as a characteristic case.We end by showing that some such consequences hold even in ZF above. (shrink)

We address three questions raised by Cummings and Foreman regarding a model of Gitik and Sharon. We first analyze the PCF-theoretic structure of the Gitik–Sharon model, determining the extent of good and bad scales. We then classify the bad points of the bad scales existing in both the Gitik–Sharon model and other models containing bad scales. Finally, we investigate the ideal of subsets of singular cardinals of countable cofinality carrying good scales.