Narrow Results

In this paper, we improve Galvin’s Theorem for ultrafilters which are $p$-point limits of $p$-points. This implies that in all the canonical inner models up to a superstrong cardinal, every $\kappa$-complete ultrafilter over a measurable cardinal $\kappa$ satisfies the Galvin property. On the other hand, we prove that supercompact cardinals always carry non-Galvin $\kappa$-complete ultrafilters. Finally, we prove that $♢(\kappa )$ implies the existence of a $\kappa$-complete filter which extends the club filter and fails to satisfy the Galvin property. This answers questions [8, Question 5.22], [4, Question 3.4] and questions, [7, Question 4.5], [6, Question 2.26].

We isolate a large class of self-adjoint operators H whose essential spectrum is determined by their behavior at x ~ ∞ and we give a canonical representation of σ_ess(H) in terms of spectra of limits at infinity of translations of H.

We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA₀, ATR₀, and .

The Ultrapower Axiom is a combinatorial principle concerning the structure of large cardinals that is true in all known canonical inner models of set theory. A longstanding test question for inner model theory is the equiconsistency of strongly compact and supercompact cardinals. In this paper, it is shown that under the Ultrapower Axiom, the least strongly compact cardinal is supercompact. A number of stronger results are established, setting the stage for a complete analysis of strong compactness and supercompactness under UA that will be carried out in the sequel to this paper.

We study which $\kappa$-distributive forcing notions of size $\kappa$ can be embedded into tree Prikry forcing notions with $\kappa$-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a $\kappa$-distributive forcing notion of size $\kappa$ be extended to a $\kappa$-complete ultrafilter.

We describe the prime ideals and, in particular, the maximal ideals in products $R=\prod D_{\lambda }$ of families ${(D_{\lambda })}_{\lambda \in \mathrm{\Lambda }}$ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra $\prod {𝒫}(max(D_{\lambda }))$, where $max(D_{\lambda })$ is the spectrum of maximal ideals of $D_{\lambda }$, and ${𝒫}$ denotes the power set. If every $D_{\lambda }$ is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of $R$. If every $D_{\lambda }$ is a Prüfer domain, we completely characterize all prime ideals of $R$.

We study the relationship between the existence of nonprincipal ultrafilters over ω and the failure of the automatic continuity, Steinhaus and Bergman properties for infinite products of finite groups.