Diamond principle

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

There are several equivalent forms of the diamond principle. One states that there is a countable collection A_α 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 ∈ A_α and C ∩ α ∈ A_α. Notice that, a weaken form which states that, there exist sets A_α ⊆ α for α < ω₁ such that for any subset A of ω₁ there is at least one infinite α with A ∩ α = A_α , is equivalent to the Continuum Hypothesis.

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 ⟨A_α : α ∈ S⟩ such that

• each A_α ⊆ α
• for every A ⊆ κ, {α ∈ S : A ∩ α = A_α} 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 A_α 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 ∈ A_α and C ∩ α ∈ A_α.

Summarize Timeline Fact Check

Jensen showed that the diamond principle ${\displaystyle \Diamond }$ implies the existence of Suslin trees. He also showed that the axiom of constructibility implies the stronger diamond-plus principle ${\displaystyle \Diamond ^{+}}$, which implies the diamond principle, which implies the continuum hypothesis.^[1] The diamond principle does not imply the existence of a Kurepa tree, but ${\displaystyle \Diamond ^{+}}$ does. Both ${\displaystyle \Diamond }$ and ${\displaystyle \Diamond ^{+}}$ are independent of the axioms of ZFC. Also, the club principle ♣ and the continuum hypothesis CH together imply ${\displaystyle \Diamond }$. However, there exist models of ♣ + ¬ CH, so ${\displaystyle \Diamond }$ and ♣ are not equivalent, rather, ♣ is weaker than ${\displaystyle \Diamond }$.^[2]

Matet proved the related principle ${\displaystyle \Diamond _{\kappa }}$, equivalent to a property of partitions of ${\displaystyle \kappa }$ with diagonal intersection of initial segments of the partitions stationary in ${\displaystyle \kappa }$.^{[3][clarification needed]}

Akemann and Weaver used ${\displaystyle \Diamond }$ to construct a C*-algebra serving as a counterexample to Naimark's problem.^[4]

For all cardinals ${\displaystyle \kappa }$ and stationary subsets ${\displaystyle S\subseteq \kappa ^{+}}$, ◊_S holds in the constructible universe. Shelah proved that for ${\displaystyle \kappa >\aleph _{0}}$, ${\displaystyle \Diamond _{\kappa ^{+}}(S)}$ follows from ${\displaystyle 2^{\kappa }=\kappa ^{+}}$ for stationary ${\displaystyle S}$ that do not contain ordinals of cofinality ${\displaystyle \kappa }$.^[5] He also showed that the diamond principle solves the Whitehead problem by implying that every Whitehead group is free.

• List of statements independent of ZFC
• Statements true in L