Английская Википедия:Diamond principle
In mathematics, and particularly in axiomatic set theory, the diamond principle Шаблон:Math is a combinatorial principle introduced by Ronald Jensen in Шаблон:Harvtxt that holds in the constructible universe (Шаблон:Math) and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the axiom of constructibility (Шаблон:Math) implies the existence of a Suslin tree.
Definitions
The diamond principle Шаблон:Math says that there exists a Шаблон:Vanchor, a family of sets Шаблон:Math for Шаблон:Math such that for any subset Шаблон:Math of ω1 the set of Шаблон:Math with Шаблон:Math is stationary in Шаблон:Math.
There are several equivalent forms of the diamond principle. One states that there is a countable collection Шаблон:Math of subsets of Шаблон:Math for each countable ordinal Шаблон:Math such that for any subset Шаблон:Math of Шаблон:Math there is a stationary subset Шаблон:Math of Шаблон:Math such that for all Шаблон:Math in Шаблон:Math we have Шаблон:Math and Шаблон:Math. Another equivalent form states that there exist sets Шаблон:Math for Шаблон:Math such that for any subset Шаблон:Mvar of Шаблон:Mvar there is at least one infinite Шаблон:Mvar with Шаблон:Mvar.
More generally, for a given cardinal number Шаблон:Math and a stationary set Шаблон:Math, the statement Шаблон:Math (sometimes written Шаблон:Math or Шаблон:Math) is the statement that there is a sequence Шаблон:Math such that
- each Шаблон:Math
- for every Шаблон:Math, Шаблон:Math is stationary in Шаблон:Math
The principle Шаблон:Math is the same as Шаблон:Math.
The diamond-plus principle Шаблон:Math states that there exists a Шаблон:Math-sequence, in other words a countable collection Шаблон:Math of subsets of Шаблон:Math for each countable ordinal α such that for any subset Шаблон:Math of Шаблон:Math there is a closed unbounded subset Шаблон:Math of Шаблон:Math such that for all Шаблон:Math in Шаблон:Math we have Шаблон:Math and Шаблон:Math.
Properties and use
Шаблон:Harvtxt showed that the diamond principle Шаблон:Math implies the existence of Suslin trees. He also showed that Шаблон: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 Шаблон:Math implies Шаблон:Math, but Shelah gave models of Шаблон:Math, so Шаблон:Math and Шаблон:Math are not equivalent (rather, Шаблон:Math is weaker than Шаблон: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>.[1]
The diamond principle Шаблон:Math does not imply the existence of a Kurepa tree, but the stronger Шаблон:Math principle implies both the Шаблон:Math principle and the existence of a Kurepa tree.
Шаблон:Harvtxt used Шаблон:Math to construct a [[C*-algebra|Шаблон:Math-algebra]] serving as a counterexample to Naimark's problem.
For all cardinals Шаблон:Math and stationary subsets Шаблон:Math, Шаблон:Math holds in the constructible universe. Шаблон:Harvtxt proved that for Шаблон:Math, Шаблон:Math follows from Шаблон:Math for stationary Шаблон:Math that do not contain ordinals of cofinality Шаблон: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 Шаблон:Mvar]]
References
Citations
- ↑ P. Matet, "On diamond sequences". Fundamenta Mathematicae vol. 131, iss. 1, pp.35--44 (1988)
