Diamondsuit

From Wikipedia, the free encyclopedia

In mathematics, and particularly in axiomatic set theory, $\Diamond_\kappa (S)$ (diamondsuit or diamond) is a certain family of combinatorial principles.

1 Definition
2 Properties and use
3 References
4 See also

[edit] Definition

For a given cardinal number $κ$ and a stationary set $S\subseteq\kappa$ , the statement $\Diamond_\kappa (S)$ 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 $κ$

When $S = κ$ , $\Diamond_\kappa (S)$ is written $\Diamond_\kappa$ , and $\Diamond_{\omega_1}$ is written $\Diamond$

[edit] Properties and use

It can be shown that ◊ ⇒ CH; also, ♣ + CH ⇒ ◊, but there also exist models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).

Charles Akemann and Nik Weaver used ◊ to construct a C^*-algebra serving as a counterexample to Naimark's problem.

For all cardinals $κ$ and stationary subsets $S \subseteq \kappa^+$ , $\Diamond_{\kappa^+} (S)$ holds in the constructible universe. Recently Shelah proved that for $\kappa>\aleph_0$ , $\diamondsuit_{\kappa^+}$ follows from $2 κ = κ +$ .