A polyadic space is a Hausdorff continuous image of some power of the onepoint compactification of a discrete space. We prove a Ramsey-like property for polyadic spaces which for Boolean spaces can be stated as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint. One corollary is that (ακ) is not a universal preimage for uniform ...

Let κ, λ be regular uncountable cardinals such that λ > κ+ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with s(κ) = λ starting from a ground model in which o(κ) = λ and prove that assuming ¬0¶, s(κ) = λ implies that o(κ) ≥ λ in the core model.

We say that κ is μ-hypermeasurable (or μ-strong) for a cardinal μ ≥ κ+ if there is an embedding j : V → M with critical point κ such that H(μ)V is included in M and j(κ) > μ. Such j is called a witnessing embedding. Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then th...

This paper develops the model theory of ordered structures that satisfy the analogue, REF (L), of the reflection principle of ZermeloFraenkel set theory. Here L is a language with a distinguished linear order <, and REF (L) consists of the universal closure of formulas of the form ∃x∀y1 < x · · · ∀y1 < x φ(y1, · · ·, yn) ↔ φ(y1, · · ·, yn), where φ(y1, · · ·, yn) is an L-formula, φ is the L-for...

Suppose κ is λ-supercompact witnessed by an elementary embedding j : V →M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton’s theorem: (1) ∀α α < cf(F (α)) and (2) α < β =⇒ F (α) ≤ F (β). In this article we address the question: assuming GCH, what additional assumptions are necessar...

For every regular cardinal κ there exists a simple complete Boolean algebra with κ generators.

For a ring R, denote by SpecΓ(κ,R) the κ-spectrum of the Γ-invariant of strongly uniform right R-modules. Recent realization techniques of Goodearl and Wehrung show that SpecΓ(א1 , R) is full for a suitable von Neumann regular algebra R, but the techniques do not extend to cardinals κ > א1. By a direct construction, we prove that for any field F and any regular uncountable cardinal κ there is a...

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible b...

For every regular cardinal κ there exists a simple complete Boolean algebra with κ generators.