Editing Diamond principle (section)

== Definitions ==
The diamond principle {{math|◊}} says that there exists a '''{{vanchor|◊-sequence}}''', a family of sets {{math|''A<sub>α</sub>'' ⊆ ''α''}} for {{math|''α'' < ''ω''<sub>1</sub>}} such that for any subset {{math|''A''}} of [[First uncountable ordinal|ω<sub>1</sub>]] the set of {{math|''α''}} with {{math|''A'' ∩ ''α'' {{=}} ''A<sub>α</sub>''}}  is [[Stationary set|stationary]] in {{math|''ω''<sub>1</sub>}}.

There are several equivalent forms of the diamond principle. One states that there is a countable collection {{math|'''A'''<sub>''α''</sub>}} of subsets of {{math|''α''}} for each countable ordinal {{math|''α''}} such that for any subset {{math|''A''}} of {{math|''ω''<sub>1</sub>}} there is a stationary subset {{math|''C''}} of {{math|''ω''<sub>1</sub>}} such that for all {{math|''α''}} in {{math|''C''}} we have {{math|''A'' ∩ ''α'' ∈ '''A'''<sub>''α''</sub>}} and {{math|''C'' ∩ ''α''  ∈ '''A'''<sub>''α''</sub>}}. Another equivalent form states that there exist sets {{math|''A''<sub>''α''</sub> ⊆ ''α''}} for {{math|''α'' < ''ω''<sub>1</sub>}} such that for any subset {{mvar|''A''}} of {{mvar|''ω''<sub>1</sub>}} there is at least one infinite {{mvar|''α''}} with {{mvar|''A'' ∩ ''α'' {{=}} ''A''<sub>''α''</sub>}}.

More generally, for a given [[cardinal number]] {{math|''κ''}} and a [[stationary set]] {{math|''S'' ⊆ ''κ''}}, the statement {{math|◊<sub>''S''</sub>}} (sometimes written {{math|◊(''S'')}} or {{math|◊<sub>''κ''</sub>(''S'')}}) is the statement that there is a [[sequence]] {{math|⟨''A<sub>α</sub>'' : ''α'' ∈ ''S''⟩}} such that

* each {{math|''A<sub>α</sub>'' ⊆ ''α''}}
* for every {{math|''A'' ⊆ ''κ''}}, {{math|{''α'' ∈ ''S'' : ''A'' ∩ ''α'' {{=}} ''A<sub>α</sub>''<nowiki>}</nowiki>}} is stationary in {{math|''κ''}}

The principle {{math|◊<sub>''ω''<sub>1</sub></sub>}} is the same as {{math|◊}}.

The diamond-plus principle {{math|◊<sup>+</sup>}} states that there exists a '''{{math|◊<sup>+</sup>}}-sequence''', in other words a countable collection {{math|'''A'''<sub>''α''</sub>}} of subsets of {{math|''α''}} for each countable ordinal α such that for any subset {{math|''A''}} of {{math|''ω''<sub>1</sub>}} there is a closed unbounded subset {{math|''C''}} of {{math|''ω''<sub>1</sub>}} such that for all {{math|''α''}} in {{math|''C''}} we have {{math|''A'' ∩ ''α'' ∈ '''A'''<sub>''α''</sub>}} and {{math|''C'' ∩ ''α'' ∈ '''A'''<sub>''α''</sub>}}.