Several variants of the Halpern-Läuchli Theorem for trees of uncountable height are investigated. For κ weakly compact, we prove that the various statements are all equivalent, and hence, the strong tree version holds for one tree on any weakly compact cardinal. For any finite d ≥ 2, we prove the consistency of the Halpern-Läuchli Theorem on d many normal κ-trees at a measurable cardinal κ, giv...

The Noetherian type of a space is the least κ for which the space has a κop-like base, i.e., a base in which no element has κ-many supersets. We prove some results about Noetherian types of (generalized) ordered spaces and products thereof. For example: the density of a product of not-too-many compact linear orders never exceeds its Noetherian type, with equality possible only for singular Noet...

We investigate the indestructibility properties of strongly compact cardinals in universes where strong compactness suffers from identity crisis. We construct an iterative poset that can be used to establish Kimchi-Magidor theorem from [22], i.e., that the first n strongly compact cardinals can be the first n measurable cardinals. As an application, we show that the first n strongly compact car...

We provide a lower bound for the consistency strength of the hypothesis proposed by S. Kamo: ∃κ > א0 s(κ) ≥ κ. 0. Introduction. In [3] ,[10] cardinal invariants on ω are generalized for uncountable regular cardinals and it is shown that some of their properties still hold true for their uncountable counterparts.However,the role of the splitting number s(κ) (for the definition see below) changes...

The additivity spectrum ADD(I) of an ideal I ⊂ P(I) is the set of all regular cardinals κ such that there is an increasing chain {Aα : α < κ} ⊂ I with ∪α<κAα / ∈ I. We investigate which set A of regular cardinals can be the additivity spectrum of certain ideals. Assume that I = B or I = N , where B denotes the σ-ideal generated by the compact subsets of the Baire space ωω , and N is the ideal o...

We introduce a covering conjecture and show that it holds below ADR + “Θ is regular”. We then use it to show that in the presence of mild large cardinal axioms, PFA implies that there is a transitive model containing the reals and ordinals and satisfying ADR + “Θ is regular”. The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first applic...

Say that κ’s measurability is destructible if there exists a <κ-closed forcing adding a new subset of κ which destroys κ’s measurability. For any δ, let λδ =df The least beth fixed point above δ. Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A1 = {δ < κ | δ is measurable, δ is not a limit of measurable cardinals, δ is not δ+ strongl...

We generalize ∇(A), which was introduced in [Sch∞], to larger cardinals. For a regular cardinal κ > א0 we denote by ∇κ(A) the statement that A ⊆ κ and for all regular θ > κ, {X ∈ [Lθ[A] ] <κ : X ∩ κ ∈ κ ∧ otp(X ∩ Ord) ∈ Card} is stationary in [Lθ[A] ] . It was shown in [Sch∞] that ∇א1(A) can hold in a set-generic extension of L. We here prove that ∇א2(A) can hold in a set-generic extension of L...

This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly σ-filtered Boolean algebras. We show that for every uncountable regular cardinal κ there are 2 pairwise non-isomorphic openly generated Boolean algebras of size κ > א1 provided there is an almost free non-free abelian group of size κ. The openly generated Boolea...

Let κ < λ be regular cardinals. We say that an embedding j : V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V . Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ++-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ++-tall measurable cardinal κ. Now more generally, suppose that κ ...