The search for diamonds

Saharon Shelah. Middle Diamond. Archive for Mathematical Logic, vol. 44 (2005), pp. 527–560. Saharon Shelah. Diamonds. Proceedings of the American Mathematical Society, vol. 138 no. 6 (2010), pp. 2151–2161. Martin Zeman. Diamond, GCH and Weak Square. Proceedings of the American Mathematical Society, vol. 138 no. 5 (2010), pp. 1853–1859. The Continuum Hypothesis (CH) implies, and is in fact equivalent to, the existence of an enumeration {Aα : α < ω1} of all bounded subsets of ω1. Given a set A ⊆ ω1, and an ordinal α < ω1, we say that the above enumeration predicts A at α, if A ∩ α = Aα. Jensen’s diamond principle, ♦, is the strengthening of CH asserting the existence of an enumeration {Aα : α < ω1} of all bounded subsets of ω1 such that every subset A ⊆ ω1 gets predicted at some limit α < ω1. Now, consider the following generalization. Given a stationary set S ⊆ κ, ♦S asserts the existence of an enumeration {Aα : α ∈ S} such that for every A ⊆ κ, the set {α ∈ S : Aα = A∩α} is stationary. This concept has been introduced by R. B. Jensen in the late 60’s of the last century; Jensen proved that ♦S holds in Gödel’s constructible universe for every stationary S ⊆ κ and every regular uncountable cardinal κ, and introduced the very first ♦-based construction of a complicated combinatorial object — a Souslin tree. Since then, these principles have drawn a considerable attention, and have been used to solve problems, not just in logic, but also in real analysis, group theory, and topology. Motivated by the utility of diamonds, the community began to study the validity of these principles, as well as weaker variants. Already in the 1970’s it was known that ♦ω1 is equivalent to several seemingly-weaker statements (Devlin, Kunen), that GCH+¬♦ω1 is consistent (Jensen), that GCH+♦ω1 +¬♦S for some stationary S ⊆ ω1 is consistent (Shelah), that ♦κ holds for every measurable cardinal κ (Kunen), that ♦2ω holds if 2 is real-valued measurable (Ketonen), and that GCH implies ♦κ+ for every uncountable cardinal κ (Jensen, Gregory, Shelah). In the 1980’s, Woodin established the consistency of ¬♦κ for a Mahlo cardinal κ, while Shelah, dealing with successor cardinals, established the consistency of GCH+¬♦Tκ for a regular uncountable κ and Tκ := {α < κ + : cf(α) = cf(κ)}, as well as, GCH+¬♦S for a singular cardinal κ and a non-reflecting stationary subset S ⊆ Tκ. In the 1990’s, Hauser, improving an earlier result of Woodin, showed that ♦Reg(κ) may consistently fail for indescribable cardinals κ, where Reg(κ) := {α < κ : cf(α) = α}, and Shelah proved that 2 = κ entails ♦κ+ for every cardinal κ ≥ iω. 1