Definition 1.

Let Sκ be a stationary set. Then the combinatorial principle S holds if and only if there is a sequence AααS such that each Aαα and for any Aκ, {αSAα=Aα} is stationary.

To get some sense of what this means, observe that for any λ<κ, {λ}κ, so the set of Aα={λ} is stationary (in κ). More strongly, suppose κ>λ. Then any subset of Tλ is bounded in κ so Aα=T on a stationary set. Since |S|=κ, it follows that 2λκ. Hence 1, the most common form (often written as just ), implies CH.

C. Akemann and N. Weaver used to construct a C*-algebra serving as a counterexample to Naimark’s problem.


  • 1 Akemann, C., and N. Weaver, Consistency of a counterexample to Naimark’s problem. Preprint available on the arXiv at
Canonical name Diamond
Date of creation 2013-03-22 12:53:49
Last modified on 2013-03-22 12:53:49
Owner Henry (455)
Last modified by Henry (455)
Numerical id 8
Author Henry (455)
Entry type Definition
Classification msc 03E65
Synonym diamond
Related topic Clubsuit
Related topic DiamondIsEquivalentToClubsuitAndContinuumHypothesis
Related topic ProofOfDiamondIsEquivalentToClubsuitAndContinuumHypothesis
Related topic CombinatorialPrinciple
Defines S