Regularity properties of ideals and ultrafilters

  title={Regularity properties of ideals and ultrafilters},
  author={Alan D. Taylor},
  journal={Annals of Mathematical Logic},
  • Alan D. Taylor
  • Published 1 May 1979
  • Mathematics
  • Annals of Mathematical Logic

Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II

We prove, assuming the existence of a huge cardinal, the consistency of fully non-regular ultrafilters on the successor of any regular cardinal. We also construct ultrafilters with ultraproducts of

Nonregular ideals

Most of the regularity properties of ideals introduced by Taylor are equivalent at successor cardinals. For $\kappa = \mu^+$ with $\mathrm{cf}(\mu)$ uncountable, we can rid the universe of dense

The nonstationary ideal in the ℙmax extension

  • P. Larson
  • Mathematics
    Journal of Symbolic Logic
  • 2007
The Boolean algebra induced by the nonstationary ideal on ω1 in this model is studied and it is shown that the induced quotient does not have a simply definable form.

Separating ultrafilters on uncountable cardinals

A uniform ultrafilterU on κ is said to be λ-separating if distinct elements of the ultrapower never projectU to the same uniform ultrafilterV on λ. It is shown that, in the presence of CH, an

The nonstationary ideal in the P max extension

The forcing construction Pmax, invented by W. Hugh Woodin, produces a model whose collection of subsets of ω1 is in some sense maximal. In this paper we study the Boolean algebra induced by the

Ideals and Generic Elementary Embeddings

This chapter covers the technique of generic elementary embeddings. These embeddings are closely analogous to conventional large cardinal embeddings, the difference being that they are definable in

Almost disjoint refinement of families of subsets of

Without any set-theoretic assumptions, we prove that every uniform ultrafilter on the set N of all natural numbers has a Comfort system, that is, an almost disjoint refinement. Moreover, we describe

Distributive ideals and partition relations

It is a theorem of Rowbottom [12] that if κ is measurable and I is a normal prime ideal on κ, then for each λ < κ, In this paper a natural structural property of ideals, distributivity, is considered


We consider saturation properties of ideals in models obtained by forcing with countable chain condition partial orderings. As sample results, we mention the following. If M[G] is obtained from a



On splitting stationary subsets of large cardinals

Abstract Let κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ+-saturated, i.e.,

Kurepa's Hypothesis and a problem of Ulam on families of measures

We prove that if Kurepa's Hypothesis holds, then on a set of cardinality ℵ1, there does not exist a family of ℵ1 non-trivial measures such that each subset is measurable with respect to at least one

Regularity of ultrafilters

If there arek++ eventually functions fromk+ intok or if there arek++ eventually different functions fromk+ then uniform ultrafilters onk+ are (k, k+)-regular.

Iterated Cohen extensions and Souslin's problem*

We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a countable dense

Some remarks on set theory

Let there be given n ordinals a,, a2, , * , a,n. It is well known that every ordinal can be written uniquely as the sum of indecomposable ordinals. (An ordinal is said to be indecomposable if it is