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Matsubara–Usuba [13] introduced the notion of skinniness and its variants for subsets of P κ λ and showed that the existence of skinny stationary subsets of P κ λ is related to large cardinal properties of ideals over P κ λ and to Jensen's diamond principle on λ. In this paper, we study the existence of skinny stationary sets in more detail.

We show that if CH holds and either (i) there exists an ω 1 -Kurepa tree, or (ii) □ ( ω 2 ) holds, then there are regular T 1 Lindelof spaces X 0 and X 1 with points G δ such that the extent of X 0 × X 1 is strictly greater than 2 ω .

We construct a normal countably tight T 1 space X with t ( X δ ) > 2 ω . This is an answer to the question posed by Dow-Juhasz-Soukup-Szentmiklossy-Weiss [5] . We also show that if the continuum is not so large, then the tightness of G δ -modifications of countably tight spaces can be arbitrary large up to the least ω 1 -strongly compact cardinal.

We study the subtlety of a cardinal κ and of ℘κλ. We show that, under a certain large cardinal assumption, it is consistent that ℘κλ is subtle for some λ>κ 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 ℘κλ can be characterized by a certain partition relation on ℘κλ. We also study the property of faintness of κ, and the subtlety of ℘κλ with the strong inclusion.

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.

A transitive model [Formula: see text] of ZFC is called a ground if the universe [Formula: see text] is a set forcing extension of [Formula: see text]. We show that the grounds of[Formula: see text][Formula: see text][Formula: see text] are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) [Formula: see text] 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.

Starting from a model with a weakly compact cardinal, we construct a model in which the weak stationary reflection principle for $\omega_2$ holds but the Fodor-type reflection principle for $\omega_2$ fails. So the stationary reflection principle for $\omega_2$ fails in this model. We also construct a model in which the semi-stationary reflection principle holds but the Fodor-type reflection principle for $\omega_2$ fails.

We study various partition properties on P κ ( λ ) . Our main result asserts that if λ λ κ = λ κ , then ( p ( NSh κ , λ κ ) ) + → ( NSS κ , λ + ) 2 , where p : P κ ( λ κ ) → P κ ( λ ) is defined by p ( x ) = x ∩ λ .

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, \Sigma_n-reflecting cardinals, \Sigma_n-correct cardinals and \Sigma_n-extendible cardinals (all for n>2) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if \kappa\ exhibits any of them, with corresponding target \theta, then in any forcing extension arising from nontrivial strategically, Comment: 19 pages. Commentary concerning this article can be made at http://jdh.hamkins.org/superstrong-never-indestructible. Minor changes in v2

Let κ be a regular uncountable cardinal, and λ be a cardinal greater than κ. Our main result asserts that if (λ< κ)< κ) = λ< κ, then (pκ, λ(NInκ, λ< κ))+ $\longrightarrow$ ((NSκ, λ[λ]< κ)+, NSκ, λs+)3 and (pκ, λ(NAInκ, λ< κ))+ $\longrightarrow$ (NSκ, λs+)3, where NSκ, λs (respectively, NSκ, λ[λ]< κ) denotes the smallest seminormal (respectively, strongly normal) ideal on Pκ (λ), NInκ, λ< κ (respectively, NAInκ, λ< κ) denotes the ideal of non-ineffable (respectively, non-almost ineffable) subsets of Pκ(λ< κ), and pκ, λ: Pκ(λ< κ) → Pκ (λ) is defined by pκ, λ(x) = x ∩ λ.