Diamond principle

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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 (L) 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.

Definitions

The diamond principle says that there exists a ◊-sequence, in other words sets Aαα for α < ω1 such that for any subset A of ω1 the set of α with Aα = 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 AαAα and CαAα. Another equivalent form states that there exist sets Aαα for α < ω1 such that for any subset A of ω1 there is at least one infinite α with Aα = Aα.

More generally, for a given cardinal number κ and a stationary set Sκ, the statement S (sometimes written ◊(S) or κ(S)) is the statement that there is a sequence Aα : αS such that

• each Aαα
• for every Aκ, {αS : Aα = 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 AαAα and CαAα.

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 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.

Akemann & Weaver (2004) used to construct a C*-algebra serving as a counterexample to Naimark's problem.

For all cardinals κ and stationary subsets Sκ+, S holds in the constructible universe. Shelah (2010) 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.