We continue the study of adequate sets which we began in (Krueger in Forcing with adequate sets of models as side conditions) by introducing the idea of a strongly adequate set, which has an additional requirement on the overlap of two models past their comparison point. We present a forcing poset for adding a club to a fat stationary subset of ω 2 with finite conditions, thereby showing that a version of the forcing posets of Friedman (Set theory: Centre de (...) Recerca Matemàtica, Barcelona, 2003–2004, Trends in Mathematics. Birkhäuser Verlag, 2006) and Mitchell (Trans Am Math Soc 361(2):561601, 2009) for adding a club on ω 2 can be developed in the context of adequate sets. (shrink)

We present a general framework for forcing on ω2 with finite conditions using countable models as side conditions. This framework is based on a method of comparing countable models as being membership related up to a large initial segment. We give several examples of this type of forcing, including adding a function on ω2, adding a nonreflecting stationary subset of, and adding an ω1‐Kurepa tree.

We develop a general framework for forcing with coherent adequate sets on [Formula: see text] as side conditions, where [Formula: see text] is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent adequate type forcings. The main theorem of the paper is that any coherent adequate type forcing preserves CH. We show that there exists a forcing poset for adding a club subset of [Formula: see text] with finite conditions while preserving CH, solving (...) a problem of Friedman [Forcing with finite conditions, in Set Theory: Centre de Recerca Matemática, Barcelona, 2003–2004, Trends in Mathematics, pp. 285–295.]. (shrink)

We define a forcing poset which adds a club subset of a given fat stationary set S⊆ω2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S \subseteq \omega_2}$$\end{document} with finite conditions, using S-adequate sets of models as side conditions. This construction, together with the general amalgamation results concerning S-adequate sets on which it is based, is substantially shorter and simpler than our original version in Krueger :119–136, 2014).

Let κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ + -saturated, i.e., are there κ + stationary subsets of κ with pairwise intersections nonstationary? Our first observation is: Theorem. NS is κ + -saturated iff for every normal ideal J on κ there is a stationary set $A \subseteq \kappa$ such that $J = NS \mid A = \{X \subseteq (...) \kappa:X \cap A \in NS\}$ . Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define "greatly Mahlo" cardinals and obtain the following: Theorem. If κ is greatly Mahlo then NS is not κ + -saturated. Theorem. If κ is ordinal Π 1 1 -indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries a κ-saturated ideal, then κ is greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π 1 1 -indescribable cardinal. These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ + -saturated. (shrink)

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)