Diamond principle

In mathematics, and particularly in axiomatic set theory, the diamond principle $◊$ is a combinatorial principle introduced by Ronald Jensen in 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.

Definitions
The diamond principle $◊$ says that there exists a ◊-sequence, a family of sets $A_{α} ⊆ α$ for $α < ω_{1}$ such that for any subset $A$ of ω1 the set of $α$ with $A ∩ α = A_{α}$ is stationary in $ω_{1}$.

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 $ω_{1}$ there is a stationary subset $C$ of $ω_{1}$ 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 $α < ω_{1}$ such that for any subset $A$ of $ω_{1}$ there is at least one infinite $α$ with $A ∩ α = A_{α}$.

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 $◊_{ω_{1}}|undefined$ 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 $ω_{1}$ there is a closed unbounded subset $C$ of $ω_{1}$ such that for all $α$ in $C$ we have $A ∩ α ∈ A_{α}$ and $C ∩ α ∈ A_{α}$.

Properties and use
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 $$\diamondsuit_\kappa$$ equivalent to a property of partitions of $$\kappa$$ with diagonal intersection of initial segments of the partitions stationary in $$\kappa$$.

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.

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. proved that for $κ > ℵ_{0}$, $◊_{κ^{+}}(S)|undefined$ follows from $2^{κ} = κ^{+}$ 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.