Diamond principle

In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by Björn Jensen (1972) that holds in the constructible universe and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that V=L implies the existence of a Suslin tree.

Definition

The diamond principle ◊ says that there exists a ◊-sequence, in other words sets A_α⊆α for α<ω₁ such that for any subset A of ω₁ the set of α with A∩α = A_α is stationary in ω₁.

More generally, for a given cardinal number $\kappa$ and a stationary set $S\subseteq \kappa$, the statement ◊_S (sometimes written ◊(S) or ◊_κ(S)) is the statement that there is a sequence $\langle A_{\alpha }:\alpha \in S\rangle$ such that

• each $A_{\alpha }\subseteq \alpha$
• for every $A\subseteq \kappa ,\{\alpha \in S:A\cap \alpha =A_{\alpha }\}$ is stationary in $\kappa$

The principle ◊_ω1 is the same as ◊.

Properties and use

Jensen (1972) showed that the diamond principle ◊ implies the existence of Suslin trees. He also showed that ◊ implies the CH. Also ♣ + CH implies ◊, but Shelah gave models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).

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

For all cardinals κ and stationary subsets S⊆κ⁺, ◊_S holds in the constructible universe. Recently Shelah proved that for κ>ℵ₀, ◊_κ⁺ follows from $2^{\kappa }=\kappa ^{+}$.

See also

• Statements true in L
• Axiom of constructibility

References

• Akemann, Charles; Weaver, Nik (2004), "Consistency of a counterexample to Naimark's problem", Proceedings of the National Academy of Sciences of the United States of America, 101 (20): 7522–7525, arXiv:math.OA/0312135, doi:10.1073/pnas.0401489101, MR 2057719
• Jensen, R. Björn (1972), "The fine structure of the constructible hierarchy", Annals of Mathematical lLogic, 4: 229–308, doi:10.1016/0003-4843(72)90001-0, MR 0309729
• Assaf Rinot, Jensen's diamond principle and its relatives, online
• S. Shelah: Whitehead groups may not be free, even assuming CH, II, Israel J. Math., 35(1980), 257–285.