On partitions into stationary sets

Abstract

We shall apply some of the results of Jensen [4] to deduce new combinatorial consequences of the axiom of constructibility, V = L. We shall show, among other things, that if V = L then for each cardinal λ there is a set A ⊆ λ such that neither A nor λ – A contain a closed set of type ω1. This is an extension of a result of Silver who proved it for λ = ω2, providing a partial answer to Problem 68 of Friedman [2].The main results of this paper were obtained independently by both authors.If λ is an ordinal, E is said to be Mahlo (or stationary) in λ, if λ – E does not contain a closed cofinal subset of λ.Consider the statements:(J1) There is a class E of limit ordinals and a sequence Cλ defined on singular limit ordinals λ such that(i) E ⋂ μ is Mahlo in μ for all regular > ω;(ii) Cλ is closed and unbounded in λ;(iii) if γ < is a limit point of Cλ, then γ is singular, γ ∉ E and Cγ = γ ⋂ Cλ.For each infinite cardinal κ:(J2,κ) There is a set E ⊂ κ+ and a sequence Cλ(Lim(λ), λ < κ+) such that(i) E is Mahlo in κ+;(ii) Cλ is closed and unbounded in λ;(iii) if cf(λ) < κ, then card Cλ < κ;(iv) if γ < λ is a limit point of Cλ then γ ∉ E and Cγ = ϣ ⋂ Cλ.