Diamond principle

{{Short description|Combinatorial principle}} In mathematics, and particularly in axiomatic set theory, the '''diamond principle''' <math>\Diamond</math> is a combinatorial principle introduced by Ronald Jensen in {{harvtxt|Jensen|1972}} 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.

== Definitions == The diamond principle {{math|◊}} says that there exists a '''{{vanchor|◊-sequence}}''', a family of sets {{math|''A_α'' ⊆ ''α''}} for {{math|''α'' < ''ω''₁}} such that for any subset {{math|''A''}} of ω₁ the set of {{math|''α''}} with {{math|''A'' ∩ ''α'' {{=}} ''A_α''}} is stationary in {{math|''ω''₁}}.

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

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

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

The principle {{math|◊_''ω''₁}} is the same as {{math|◊}}.

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

== Properties and use == {{harvtxt|Jensen|1972}} showed that the diamond principle {{math|◊}} implies the existence of Suslin trees. He also showed that {{math|''V'' {{=}} ''L''}} implies the diamond-plus principle, which implies the diamond principle, which implies CH. In particular the diamond principle and the diamond-plus principle are both independent of the axioms of ZFC. Also {{math|♣ + CH}} implies {{math|◊}}, but 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|◊⁺}} principle implies both the {{math|◊}} principle and the existence of a Kurepa tree.

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

For all cardinals {{math|''κ''}} and stationary subsets {{math|''S'' ⊆ ''κ''⁺}}, {{math|◊_''S''}} holds in the constructible universe. {{harvtxt|Shelah|2010}} proved that for {{math|''κ'' > ℵ₀}}, {{math|◊_''κ''⁺(''S'')}} follows from {{math|2^''κ'' {{=}} ''κ''⁺}} 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 group is free.

== See also == * List of statements independent of ZFC * 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