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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“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)