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 .
Read more about this topic: Diamond Principle
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“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)