Diamond principle

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

Definitions

The diamond principle Template:Math says that there exists a Template:Vanchor, 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. Another equivalent form states that there exist sets Template:Math for Template:Math such that for any subset Template:Mvar of Template:Mvar there is at least one infinite Template:Mvar with Template:Mvar.

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

Template:Harvtxt showed that the diamond principle Template:Math implies the existence of Suslin trees. He also showed that Template:Math 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 Template:Math implies Template:Math, but Shelah gave models of Template:Math, so Template:Math and Template:Math are not equivalent (rather, Template:Math is weaker than Template:Math).

Matet proved the principle <math>\diamondsuit_\kappa</math> equivalent to a property of partitions of <math>\kappa</math> with diagonal intersection of initial segments of the partitions stationary in <math>\kappa</math>.<ref>P. Matet, "On diamond sequences". Fundamenta Mathematicae vol. 131, iss. 1, pp.35--44 (1988)</ref>

The diamond principle Template:Math does not imply the existence of a Kurepa tree, but the stronger Template:Math principle implies both the Template:Math principle and the existence of a Kurepa tree.

Template:Harvtxt used Template:Math to construct a [[C*-algebra|Template:Math-algebra]] serving as a counterexample to Naimark's problem.

For all cardinals Template:Math and stationary subsets Template:Math, Template:Math holds in the constructible universe. Template:Harvtxt proved that for Template:Math, Template:Math follows from Template:Math for stationary Template:Math that do not contain ordinals of cofinality Template:Math.

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

See also

• List of statements independent of ZFC
• [[Statements true in L|Statements true in Template:Mvar]]

References

• Template:Cite journal
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• Template:Cite book
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