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<λMα. 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.