Survey on the Tukey theory of ultrafilters

Natasha Dobrinen

Home/Archive/arXiv

arXiv · 1401.8108

Survey on the Tukey theory of ultrafilters

Natasha Dobrinen

View original source ↗Read source document ↗Export metadata ↓

Abstract

This article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell's Problem and the question of how closely related the Rudin-Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guarantee that an ultrafilter is below the Tukey maximum. The known canonical forms for cofinal maps on ultrafilters are reviewed, as well as their applications to finding which structures embed into the Tukey types of ultrafilters. With the addition of some Ramsey theory, fine analyses of the structures at the bottom of the Tukey hierarchy are made.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Natasha Dobrinen. 2014-01-31. Survey on the Tukey theory of ultrafilters. https://arxiv.org/abs/1401.8108

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Forcing with Symmetric Systems of Models of Two Types

The purpose of this paper is to present a general method for forcing on $ω_2$ and $ω_3$ with finite conditions, while preserving all cardinals and some fragments of $\mathrm{GCH}$. This method is based on the technique of forcing with finite symmetric systems of elementary submodels, and improves earlier versions of this forcing by including models of two types. We will present several applications of the pure side condition forcing and variants thereof, by adding a Kurepa tree on $ω_2$, a club subset of $ω_2$ that avoids infinite sets from the ground model, a function bounding every canonical function below $ω_3$ on a club, and a simplified $(ω_2,1)$-morass.

math.LO↗

arXiv2026

Compactness beyond choice and HOD

We characterise exacting, rank-Berkeley, Berkeley, and club Berkeley cardinals by compactness properties of logics and, equivalently, by compactness properties of certain topological spaces. This raises the strength known to be obtainable by statements about compactness into the realm of choiceless large cardinal axioms, the strongest known statements in terms of consistency strength. Moreover, it shows that compactness assumptions can directly violate the axiom of choice and the axiom $V=\text{HOD}$.

math.LO↗

arXiv2026

Two applications of the point-free coderivative

We present two new applications of Simmons' point-free Cantor-Bendixson coderivative operator in intuitionistic logic. First, we use it to give a simplified proof of the recent result of Xu and Ye that the free Heyting algebra on two generators does not occur as the Heyting algebra of subterminal objects in any elementary topos. Then we use it to prove that complete Heyting algebra semantics is not strongly complete for intuitionistic second-order propositional logic: semantic consequence from an arbitrary set of assumptions does not coincide with ordinary syntactic consequence.

math.LO↗