Diamond principle

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 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.

Definitions

The diamond principle ◊ says that there exists a ◊-sequence, in other words 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. Another equivalent form 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.

More generally, for a given cardinal number $\kappa$ and a stationary set $S\subseteq\kappa$, the statement ◊_S (sometimes written ◊(S) or ◊_κ(S)) is the statement that there is a sequence $\langle A_\alpha: \alpha \in S \rangle$ such that

• each $A_\alpha \subseteq \alpha$
• for every $A \subseteq \kappa, \{\alpha \in S: A \cap \alpha = A_\alpha\}$ is stationary in $\kappa$

The principle ◊_ω1 is the same as ◊.

The diamond plus principle ◊⁺ says 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.

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 the 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 ◊).

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. Recently Shelah proved that for κ>ℵ₀, ◊_κ⁺(S) follows from $2^\kappa=\kappa^+$ for stationary S that do not contain ordinals of cofinality $\kappa$.

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

See also

• Statements true in L