Diamond (mathematics)

In mathematics, and particularly in axiomatic set theory, $Diamond\_kappa\; (S)$ (diamond) is a certain family of combinatorial principles.

Definition

For a given cardinal number $kappa$ and a stationary set $Ssubseteqkappa$, the statement $Diamond\_kappa\; (S)$ is the statement that there is a sequence $langle\; A\_alpha:\; alpha\; in\; S\; angle$ such that

* each $A\_alpha\; subseteq\; alpha$
* for every $A\; subseteq\; kappa,\; \{alpha\; in\; S:\; A\; cap\; alpha\; =\; A\_alpha\}$ is stationary in $kappa$

When $S\; =\; kappa$, $Diamond\_kappa\; (S)$ is written $Diamond\_kappa$, and $Diamond\_\{omega\_1\}$ is written $Diamond$

Properties and use

Diamond principles are classically used to build Suslin trees.

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 $kappa$ 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 $2^kappa=kappa^+$.

References

* Charles Akemann, Nik Weaver, "Consistency of a counterexample to Naimark's problem", [http://arxiv.org/abs/math.OA/0312135 online]

ee also

*statements true in L
*axiom of constructibility