Todd Eisworth

Successors of Singular Cardinals

Successors of singular cardinals are a peculiar—although they are successor cardinals, they can still exhibit some of the behaviors typically associated with large cardinals. In this chapter, we examine the combinatorics of successors of singular cardinals in detail. We use stationary reflection as our point of entry into the subject, and we sketch Magidor’s proof that it is consistent that all stationary subsets of such a cardinal reflect. Further consideration of Magidor’s proof brings us to Shelah’s ideal I[λ] and the related Approachability Property (AP); we give a fairly comprehensive treatment of these topics. Building on this, we then turn to squares, scales, and the influence these objects exert on questions of pertaining to reflection phenomena. The chapter concludes with a brief look at square-brackets partition relations and their relation to club-guessing principles.

CH with no Ostaszewski spaces

This is the published version, also available here: http://dx.doi.org/10.1090/S0002-9947-99-02407-1. First published in Trans. Amer. Math. Soc. in 1999, published by the American Mathematical Society.

Antidiamond principles and topological applications

We investigate some combinatorial statements that are strong enough to imply that ? fails (hence the name antidiamonds); yet most of them are also compatible with CH. We prove that these axioms have many consequences in set-theoretic topology, including the consistency, modulo large cardinals, of a Yes answer to a problem on linearly Lindelof spaces posed by Arhangelskiĭ and Buzyakova (1998).

Club-guessing, stationary reflection, and coloring theorems

Abstract We obtain very strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. In particular, the simplest of our results establishes that if μ is singular and μ + → [ μ + ] μ + 2 , then there is a regular cardinal θ μ such that any fewer than cf ( μ ) stationary subsets of S ≥ θ μ + must reflect simultaneously.

First countable, countably compact spaces and the continuum hypothesis

We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of ω 1 with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelahs iteration theorems appearing in Chapters V and VIII of Proper and improper forcing (1998), as well as Eisworth and Roitmans (1999) iteration theorem. We close the paper with a ZFC example (constructed using Shelahs club-guessing sequences) that shows similar results do not hold for closed pre-images of ω 2 .

GENTLY KILLING S{SPACES

We produce a model of ZFC in which there are no locally compact first countable S-spaces, and in which 2ℵ0<2ℵ1. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size ℵ1, answering a question of the second author [9].

Successors of singular cardinals and coloring theorems I

Abstract.We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [1], but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.

On Ideals Related to I[λ]

We describe a recipe for generating normal ideals on successors of singular cardinals. We show that these ideals are related to many weakenings of that have appeared in the literature. Our main purpose, however, is to provide an organized list of open questions related to these ideals. Throughout this note, we will let λ denote the successor of a singular cardinal μ. We will also let χ denote some regular cardinal much larger than λ; we will be concerned with elementary submodels of various expansions of 〈H(χ),∈, <χ〉, where <χ is some well–ordering of H(χ) (the sets hereditarily of cardinality < χ). Suppose M ≺ 〈H(χ),∈, <χ〉 satisfies • |M | = μ, and • M ∩ λ is an initial segment of λ. The ordinal δ := M ∩λ lies in the interval (μ, λ), so in particular δ is singular with cofinality < μ. The ideals of concern to us have to do with asking about the extent to which the singularity of δ can be witnessed by a set “covered” by M ∩ [λ]. For example, is there a set A ⊆ δ of order–type cf(δ) with every initial segment in M? Can we find such an A that is also closed and unbounded? What about if we demand only that every countable subset of A is covered by a set in M ∩ [λ]? What follows is one way to systematically generate ideals associated to such questions. Our goal in this note is merely to demonstrate that many weakenings of considered in the literature are instances of such a scheme, and to point out some fairly general questions that ought to be investigated further. Definition 1. Let λ be a regular cardinal. A λ–approximating sequence is a sequence M = 〈Mα : α < λ〉 such that (1) M is a continuous ∈–chain of elementary submodels of 〈H(χ),∈, <χ〉, (2) 〈Mj : j ≤ i〉 ∈ Mi+1, (3) λ ∈ M0, and for each α < λ, (4) |Mα| < λ, and (5) Mα ∩ λ is an initial segment of λ. A λ–approximating sequence is said to be over x if x ∈ ⋃ α<λ Mα. Our recipe for generating normal ideals will use λ–approximating sequences. Each instance of the recipe depends on two things – how we want our ordinals Date: February 2, 2005. These remarks are an organized version of questions asked during a Problem Session at the Singular Cardinal Combinatorics Workshop held at BIRS during May 2004. The author thanks the organizers inviting him to the conference, and the staff at BIRS for the wonderful job they did hosting the workshop.

Near coherence and filter games

Abstract. We investigate a two-player game involving pairs of filters on ω. Our results generalize a result of Shelah ([7] Chapter VI) dealing with applications of game theory in the study of ultrafilters.