In mathematics, and particularly in axiomatic set theory, the diamond principle is a combinatorial principle introduced by Ronald Jensen in that holds in the constructible universe and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the axiom of constructibility implies the existence of a Suslin tree.
The diamond principle says that there exists a , a family of sets for such that for any subset of ω1 the set of with is stationary in .
There are several equivalent forms of the diamond principle. One states that there is a countable collection of subsets of for each countable ordinal such that for any subset of there is a stationary subset of such that for all in we have and . Another equivalent form states that there exist sets for such that for any subset of there is at least one infinite with .
More generally, for a given cardinal number and a stationary set, the statement (sometimes written or) is the statement that there is a sequence such that