Diamond principle

Template:Short description In set theory, the diamond principle, denoted $◊$ , is a combinatorial principle introduced by Ronald Jensen that holds in the constructible universe and that implies the continuum hypothesis.Template:Sfnp Jensen extracted the diamond principle from his proof that the axiom of constructibility implies the existence of a Suslin tree.

Definitions

The diamond principle Template:Math says that there exists a Template:Vanchor; that is, a family of sets Template:Math for Template:Math such that for any subset Template:Math of ω₁ the set of Template:Math with Template:Math is stationary in Template:Math.

There are several equivalent forms of the diamond principle. One states that there is a countable collection Template:Math of subsets of Template:Math for each countable ordinal Template:Math such that for any subset Template:Math of Template:Math there is a stationary subset Template:Math of Template:Math such that for all Template:Math in Template:Math we have Template:Math and Template:Math. Notice that, a weaken form which states that, there exist sets Template:Math for Template:Math such that for any subset Template:Mvar of Template:Math there is at least one infinite Template:Mvar with Template:Mvar , is equivalent to the Continuum Hypothesis.

More generally, for a given cardinal number Template:Math and a stationary set Template:Math, the statement Template:Math (sometimes written Template:Math or Template:Math) is the statement that there is a sequence Template:Math such that

each Template:Math
for every Template:Math, Template:Math is stationary in Template:Math

The principle Template:Math is the same as Template:Math.

The diamond-plus principle Template:Math states that there exists a Template:Math-sequence, in other words a countable collection Template:Math of subsets of Template:Math for each countable ordinal α such that for any subset Template:Math of Template:Math there is a closed unbounded subset Template:Math of Template:Math such that for all Template:Math in Template:Math we have Template:Math and Template:Math.

Properties and use

Jensen showed that the diamond principle $◊$ implies the existence of Suslin trees. He also showed that the axiom of constructibility implies the stronger diamond-plus principle $◊^{+}$ , which implies the diamond principle, which implies the continuum hypothesis.Template:Sfnp The diamond principle does not imply the existence of a Kurepa tree, but $◊^{+}$ does. Both $◊$ and $◊^{+}$ are independent of the axioms of ZFC. Also, the club principle Template:Math and the continuum hypothesis Template:Math together imply $◊$ . However, there exist models of Template:Math, so $◊$ and Template:Math are not equivalent, rather, Template:Math is weaker than $◊$ .Template:Sfnp

Matet proved the related principle $◊_{κ}$ , equivalent to a property of partitions of $κ$ with diagonal intersection of initial segments of the partitions stationary in $κ$ .Template:Sfnp Template:Clarification needed

Akemann and Weaver used $◊$ to construct a [[C*-algebra|Template:Math-algebra]] serving as a counterexample to Naimark's problem.Template:Sfnp

For all cardinals $κ$ and stationary subsets $S \subseteq κ^{+}$ , Template:Math holds in the constructible universe. Shelah proved that for $κ > ℵ_{0}$ , $◊_{κ^{+}} (S)$ follows from $2^{κ} = κ^{+}$ for stationary $S$ that do not contain ordinals of cofinality $κ$ .Template:Sfnp He also showed that the diamond principle solves the Whitehead problem by implying that every Whitehead group is free.