Editing Diamond principle

In [[mathematics]], and particularly in [[axiomatic set theory]], the '''diamond principle''' {{math|◊}} is a [[combinatorial principle]] introduced by [[Ronald Jensen]] in {{harvtxt|Jensen|1972}} that holds in the [[constructible universe]] ({{math|''L''}}) and that implies the [[continuum hypothesis]]. Jensen extracted the diamond principle from his proof that the [[axiom of constructibility]] ({{math|''V'' {{=}} ''L''}}) implies the existence of a [[Suslin tree]].

== 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>}}.

== 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.

== See also ==
* [[List of statements independent of ZFC]]
* [[Statements true in L|Statements true in {{mvar|L}}]]

== References ==
{{refbegin|30em}}
*{{cite journal |last1=Akemann |first1=Charles |last2=Weaver |first2=Nik |year=2004 |title=Consistency of a counterexample to Naimark's problem |journal=[[Proceedings of the National Academy of Sciences]] |doi=10.1073/pnas.0401489101 |mr=2057719 |pmid=15131270 |pmc=419638 |arxiv=math.OA/0312135 |bibcode=2004PNAS..101.7522A |volume=101 |issue=20 |pages=7522–7525|doi-access=free }}
*{{cite journal |last1=Jensen |first1=R. Björn |year=1972 |title=The fine structure of the constructible hierarchy |journal=Annals of Mathematical Logic |doi=10.1016/0003-4843(72)90001-0 |doi-access=free |mr=0309729 |volume=4 |issue=3 |pages=229–308}}
*{{cite book |last=Rinot |first=Assaf |year=2011 |title=Set theory and its applications |chapter=Jensen's diamond principle and its relatives |series=Contemporary Mathematics |publisher=AMS |location=Providence, RI |isbn=978-0-8218-4812-8 |mr=2777747 |arxiv=0911.2151 |bibcode=2009arXiv0911.2151R |volume=533 |pages=125–156 |url=http://papers.assafrinot.com/?num=s01}}
*{{cite journal |last=Shelah |first=Saharon |author-link=Saharon Shelah |year=1974 |title=Infinite Abelian groups, Whitehead problem and some constructions |journal=[[Israel Journal of Mathematics]] |doi=10.1007/BF02757281 |doi-access= |s2cid=123351674 |mr=0357114 |volume=18 |issue=3 |pages=243–256}}
*{{cite journal |last=Shelah |first=Saharon |author-link=Saharon Shelah |year=2010 |title=Diamonds |journal=[[Proceedings of the American Mathematical Society]] |doi=10.1090/S0002-9939-10-10254-8 |doi-access=free |volume=138 |issue=6 |pages=2151–2161}}
{{refend}}
=== Citations ===
{{reflist}}

[[Category:Set theory]]
[[Category:Mathematical principles]]
[[Category:Independence results]]
[[Category:Constructible universe]]