We deal with several pcf problems: we characterize another version of exponentiation: maximal number of κ-branches in a tree with λ nodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariants for each λ with a pcf restriction and investigate further T D (f). The sections can be read independently, although there are some minor dependencies.

Abstract.We prove in ZF+DC, e.g. that: if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mu=|{\cal H}(\mu)|$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mu>\cf(\mu)>\aleph_0$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mu ^+$\end{document} is regular but non measurable. This is in contrast with the results on measurability for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mu=\aleph_\omega$\end{document} due to Apter and Magidor [ApMg].

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the failure (...) of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for manyλ ⌝ PT(λ, ℵ1). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular λ which is a limit of supercompacts the bad stationary set concentrates on the “right” cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular λ {δ<+ : cfδ<λ} is good. It is proved in ZFC that ifλ=cfλ>ℵ1 then {α<+ : cfα<λ} is the union of λ sets on which there are squares. (shrink)