In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe 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.
The diamond principle ◊ says that there exists a ◊-sequence, in other words 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.
- for every is stationary in
The principle ◊ω1 is the same as ◊.
The diamond plus principle ◊+ says 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
Jensen (1972) 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 the 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 ◊).
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.
For all cardinals κ and stationary subsets S⊆κ+, ◊S holds in the constructible universe. Recently Shelah proved that for κ>ℵ0, ◊κ+(S) follows from for stationary S that do not contain ordinals of cofinality .
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- Shelah, S. (1974), "Infinite Abelian groups, Whitehead problem and some constructions", Israel Journal of Mathematics 18 (3): 243–256, doi:10.1007/BF02757281, MR 0357114