Editing Diamond principle (section)

== Properties and use ==
{{harvtxt|Jensen|1972}} showed that the diamond principle {{math|◊}} implies the existence of [[Suslin tree]]s. He also showed that {{math|[[V=L|''V'' {{=}} ''L'']]}} implies the diamond-plus principle, which implies the diamond principle, which implies [[continuum hypothesis|CH]]. In particular the diamond principle and the diamond-plus principle are both [[Independence (mathematical logic)|independent]] of the axioms of [[ZFC]]. Also {{math|[[clubsuit|♣]] + CH}} implies {{math|◊}}, but [[Saharon Shelah|Shelah]] gave models of {{math|♣ + ¬ CH}}, so {{math|◊}} and {{math|♣}} are not equivalent (rather, {{math|♣}} is weaker than {{math|◊}}).

Matet proved the principle <math>\diamondsuit_\kappa</math> equivalent to a property of partitions of <math>\kappa</math> with diagonal intersection of initial segments of the partitions stationary in <math>\kappa</math>.<ref>P. Matet, "[https://eudml.org/doc/211737 On diamond sequences]". Fundamenta Mathematicae vol. 131, iss. 1, pp.35--44 (1988)</ref>

The diamond principle {{math|◊}} does not imply the existence of a [[Kurepa tree]], but the stronger {{math|◊<sup>+</sup>}} principle implies both the {{math|◊}} principle and the existence of a Kurepa tree.

{{harvtxt|Akemann|Weaver|2004}} used {{math|◊}} to construct a [[C*-algebra|{{math|''C''*}}-algebra]] serving as a [[counterexample]] to [[Naimark's problem]].

For all cardinals {{math|''κ''}} and [[stationary subset]]s {{math|''S'' ⊆ ''κ''<sup>+</sup>}}, {{math|◊<sub>''S''</sub>}} holds in the [[constructible universe]]. {{harvtxt|Shelah|2010}} proved that for {{math|''κ'' > ℵ<sub>0</sub>}}, {{math|◊<sub>''κ''<sup>+</sup></sub>(''S'')}} follows from {{math|2<sup>''κ''</sup> {{=}} ''κ''<sup>+</sup>}} for stationary {{math|''S''}} that do not contain ordinals of cofinality {{math|''κ''}}.

Shelah showed that the diamond principle solves the [[Whitehead problem]] by implying that every [[Whitehead problem|Whitehead group]] is free.