We introduce the notion of skinniness for subsets of $\mathcal{P}_\kappa \lambda$ and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or $2^\lambda$-saturation of $\mathrm{NS}_{\kappa\lambda}\mid X$, where $\mathrm{NS}_{\kappa\lambda}$ denotes the non-stationary ideal over $\mathcal{P}_\kappa \lambda$, implies the existence of a skinny stationary subset of $X$. We also show that if $\lambda$ is a singular cardinal, then there is no skinnier stationary subset of $\mathcal{P}_\kappa \lambda$. Furthermore, if $\lambda$ is a strong limit singular cardinal, there (...) is no skinny stationary subset of $\mathcal{P}_\kappa \lambda$. Combining these results, we show that if $\lambda$ is a strong limit singular cardinal, then $\mathrm{NS}_{\kappa\lambda}\mid X$ can satisfy neither precipitousness nor $2^\lambda$-saturation for every stationary $X \subseteq \mathcal{P}_\kappa \lambda$. We also indicate that $\diamondsuit_\lambda(E^{\lambda}_{<\kappa})$, where $E^{\lambda}_{<\kappa} \stackrel{\mathrm{def}}{=} \{\alpha < \lambda \mid \mathrm{cf}(\alpha) < \kappa\}$, is equivalent to the existence of a skinnier (or skinniest) stationary subset of $\mathcal{P}_\kappa \lambda$ under some cardinal arithmetical hypotheses. (shrink)

A transitive model M of ZFC is called a ground if the universe V is a set forcing extension of M. We show that the grounds ofV are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, the mantle, the intersection of all grounds, must be a model of ZFC. V has only set many grounds if and only if the mantle is a ground. We also show that if the universe (...) has some very large cardinal, then the mantle must be a ground. (shrink)

The mantle is the intersection of all ground models of V. We show that if there exists an extendible cardinal then the mantle is the smallest ground model of V.

For an uncountable cardinal κ, let be the assertion that every ω1‐stationary preserving poset of size is semiproper. We prove that is a strong principle which implies a strong form of Chang's conjecture. We also show that implies that is presaturated.

Let A be a non-empty set. A set $S\subseteq \mathcal{P}(A)$ is said to be stationary in $\mathcal{P}(A)$ if for every f: [A] <ω → A there exists x ∈ S such that x ≠ A and f"[x] <ω ⊆ x. In this paper we prove the following: For an uncountable cardinal λ and a stationary set S in \mathcal{P}(\lambda) , if there is a regular uncountable cardinal κ ≤ λ such that {x ∈ S: x ⋂ κ ∈ κ} is (...) stationary, then S can be split into κ disjoint stationary subsets. (shrink)

Starting with a λ-supercompact cardinal κ, where λ is a regular cardinal greater than or equal to κ, we produce a model with a stationary subset S of such that , the ideal generated by the non-stationary ideal over together with , is λ+-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis . We also show that in our model we can make -saturated, where S is the (...) set of all such that , the order type of x, is a regular cardinal and x is stationary in sup. Furthermore we construct a model where is κ+-saturated but GCH fails. We show that if SS is stationary in , then S can be split into λ many disjoint stationary subsets. (shrink)

We study the subtlety of a cardinal κ and of. We show that, under a certain large cardinal assumption, it is consistent that is subtle for some but κ is not subtle, and the consistency of such a situation is much stronger than the existence of a subtle cardinal. We show that the subtlety of can be characterized by a certain partition relation on. We also study the property of faintness of κ, and the subtlety of with the strong inclusion.

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing Q∈Vθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...) \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q} \in V_\theta}$$\end{document}, the cardinal κ will exhibit none of the large cardinal properties with target θ or larger. (shrink)

We investigate the partition property of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\kappa}\lambda}$$\end{document}. Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\kappa}\lambda}$$\end{document} carries a (λκ, 2)-distributive normal ideal without the partition property, then λ is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^1_n}$$\end{document}-indescribable for all n < ω but not \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^2_1}$$\end{document} -indescribable. (2) If cf(λ) ≥ κ, then every ineffable subset of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\kappa}\lambda}$$\end{document} has the partition property. (3) If cf(λ) ≥ κ, then the completely ineffable ideal over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\kappa}\lambda}$$\end{document} has the partition property. (shrink)

We answer a question of Usuba by showing that the combinatorial principle $\mathrm {UB}_\lambda $ can fail at a singular cardinal. Furthermore, $\lambda $ can be taken to be $\aleph _\omega.$.

We prove a variety of theorems about stationary set reflection and concepts related to internal approachability. We prove that an implication of Fuchino–Usuba relating stationary reflection to a version of Strong Chang’s Conjecture cannot be reversed; strengthen and simplify some results of Krueger about forcing axioms and approachability; and prove that some other related results of Krueger are sharp. We also adapt some ideas of Woodin to simplify and unify many arguments in the literature involving preservation of forcing axioms.

In an article published in 1974, Menas conjectured that any stationary subset of can be split in many pairwise disjoint stationary subsets. Even though the conjecture was shown long ago by Baumgartner and Taylor to be consistently false, it is still haunting papers on. In which situations does it hold? How much of it can be proven in ZFC? We start with an abridged history of the conjecture, then we formulate a new version of it, and finally we keep weakening (...) this new assertion until, building on the work of Usuba, we hit something we can prove. (shrink)

We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if, then (a) is not (λ, 2)‐distributive, and (b) does not hold.