Diamond principle
In
Definitions
The diamond principle ◊ says that there exists a ◊-sequence, a family of sets Aα ⊆ α for α < ω1 such that for any subset A of ω1 the set of α with A ∩ α = Aα is stationary in ω1.
There are several equivalent forms of the diamond principle. One states that there is a countable collection Aα 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 ∈ Aα and C ∩ α ∈ Aα. Another equivalent form states that there exist sets Aα ⊆ α for α < ω1 such that for any subset A of ω1 there is at least one infinite α with A ∩ α = Aα.
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 ⟨Aα : α ∈ S⟩ such that
- each Aα ⊆ α
- for every A ⊆ κ, {α ∈ S : A ∩ α = Aα} is stationary in κ
The principle ◊ω1 is the same as ◊.
The diamond-plus principle ◊+ states that there exists a ◊+-sequence, in other words a countable collection Aα 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 ∈ Aα and C ∩ α ∈ Aα.
Properties and use
gave models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).Matet proved the principle equivalent to a property of partitions of with diagonal intersection of initial segments of the partitions stationary in .[1]
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
proved that for κ > ℵ0, ◊κ+(S) follows from 2κ = 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.
See also
- List of statements independent of ZFC
- Statements true in L
References
- Akemann, Charles; Weaver, Nik (2004). "Consistency of a counterexample to Naimark's problem". PMID 15131270.
- Jensen, R. Björn (1972). "The fine structure of the constructible hierarchy". Annals of Mathematical Logic. 4 (3): 229–308. MR 0309729.
- Rinot, Assaf (2011). "Jensen's diamond principle and its relatives". Set theory and its applications. Contemporary Mathematics. Vol. 533. Providence, RI: AMS. pp. 125–156. MR 2777747.
- S2CID 123351674.
- .
Citations
- ^ P. Matet, "On diamond sequences". Fundamenta Mathematicae vol. 131, iss. 1, pp.35--44 (1988)