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Diamond principle

Combinatorial principle that there exists a family of sets 𝐴(𝛼)⊆𝛼 for 𝛼<ω₁ such that for any 𝐴⊆ω₁, the set of 𝛼’s with 𝐴∩𝛼=𝐴(𝛼) is stationary in ω₁

In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe (L) and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the axiom of constructibility (V = L) implies the existence of a Suslin tree.

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Definitions

The diamond principle ◊ says that there exists a ◊-sequence, a family of sets Aα ⊆ α for α < ω1 such that for any subset A of ω1 the set of α with A ∩ α = Aα is stationary in ω1.

There are several equivalent forms of the diamond principle. One states that there is a countable collection Aα of subsets of α for each countable ordinal α such that for any subset A of ω1 there is a stationary subset C of ω1 such that for all α in C we have A ∩ α ∈ Aα and C ∩ α ∈ Aα. Another equivalent form states that there exist sets Aα ⊆ α for α < ω1 such that for any subset A of ω1 there is at least one infinite α with A ∩ α = Aα.

More generally, for a given cardinal number κ and a stationary set S ⊆ κ, the statement ◊S (sometimes written ◊(S) or ◊κ(S)) is the statement that there is a sequence ⟨Aα : α ∈ S⟩ such that

each Aα ⊆ α
for every A ⊆ κ, {α ∈ S : A ∩ α = Aα} is stationary in κ

The principle ◊ω1 is the same as ◊.

The diamond-plus principle ◊+ states that there exists a ◊+-sequence, in other words a countable collection Aα of subsets of α for each countable ordinal α such that for any subset A of ω1 there is a closed unbounded subset C of ω1 such that for all α in C we have A ∩ α ∈ Aα and C ∩ α ∈ Aα.

Properties and use

Jensen (1972) showed that the diamond principle ◊ implies the existence of Suslin trees. He also showed that V = L implies the diamond-plus principle, which implies the diamond principle, which implies CH. In particular the diamond principle and the diamond-plus principle are both independent of the axioms of ZFC. Also ♣ + CH implies ◊, but Shelah gave models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).

Matet proved the principle ♢ κ {\displaystyle \diamondsuit _{\kappa }} equivalent to a property of partitions of κ {\displaystyle \kappa } with diagonal intersection of initial segments of the partitions stationary in κ {\displaystyle \kappa } .¹

The diamond principle ◊ does not imply the existence of a Kurepa tree, but the stronger ◊+ principle implies both the ◊ principle and the existence of a Kurepa tree.

Akemann & Weaver (2004) used ◊ to construct a C*-algebra serving as a counterexample to Naimark's problem.

For all cardinals κ and stationary subsets S ⊆ κ+, ◊S holds in the constructible universe. Shelah (2010) proved that for κ > ℵ0, ◊κ+(S) follows from 2κ = κ+ for stationary S that do not contain ordinals of cofinality κ.

Shelah showed that the diamond principle solves the Whitehead problem by implying that every Whitehead group is free.

References

P. Matet, "On diamond sequences". Fundamenta Mathematicae vol. 131, iss. 1, pp.35--44 (1988) https://eudml.org/doc/211737 ↩

Definitions

Properties and use

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References