Successors of Singular Cardinals

Abstract

We formulate and prove (in ZFC) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle Pr 1 (µ + , µ + , µ + , cf(µ)) for singular µ.Date: October 31, 2018.

Stationary sets added when forcing squares

“…Finally, we use a fact drawn from the theory of the approachability ideal, namely that * κ implies the approachability property at κ, which in turn implies that stationary subsets of S ⊂ κ + ∩ cof(τ ) are preserved by τ + -closed forcing posets [2]. Proof Fix a C-generic filter G and a stationary set S ⊂ κ + ∩ cof(τ ) such that S ∈ V [G].…”

Some consequences of reflection on the approachability ideal

“…Moreover in the case that κ is a strong limit and singular, I[κ + ] = I[κ + , κ] (section 3.4 and proposition 3.23 of [3]). For this reason we feel free to concentrate our attention on the notion of weak approachability which applies to a more general context.…”

“…A main result of Shelah is that there is a stationary set in I[κ + ] for any singular cardinal κ (theorem 3.18 in [3]). There are several applications of this ideal to the combinatorics of singular cardinals; we remind the reader of one of them and refer him to section 3 of [3] for a detailed account: the extent of this ideal can be used to size the large cardinal properties of κ. I[κ + , κ] is trivial unless the cardinals below κ + have very strong combinatorial properties (in the range of supercompactness). Thus for example if the square at κ holds, then I[κ + ] = I[κ + , κ] = P (κ + ) (theorem 3.13 of [3]).…”

“…There are several applications of this ideal to the combinatorics of singular cardinals; we remind the reader of one of them and refer him to section 3 of [3] for a detailed account: the extent of this ideal can be used to size the large cardinal properties of κ. I[κ + , κ] is trivial unless the cardinals below κ + have very strong combinatorial properties (in the range of supercompactness). Thus for example if the square at κ holds, then I[κ + ] = I[κ + , κ] = P (κ + ) (theorem 3.13 of [3]). On the other hand if λ is strongly compact and κ > λ is singular of cofinality θ < λ, then there is a stationary subset of κ + of points of cofinality less than λ which is not in I[κ + , κ] (Shelah, theorem 3.20 of [3]).…”

Splitting stationary sets in Pκλ for λ with small cofinality

“…Here we present some basic notations and facts from that theory. These can be found in Abraham-Magidor [2], Cummings [6], Eisworth [8], and Shelah [14].…”