diamond principle

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

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|ω₁}} there is a stationary subset {{math|C}} of {{math|ω₁}} such that for all {{math|α}} in {{math|C}} we have {{math|A ∩ α ∈ A_α}} and {{math|C ∩ α ∈ A_α}}. Another equivalent form states that there exist sets {{math|A_α ⊆ α}} for {{math|α < ω₁}} such that for any subset {{mvar|A}} of {{mvar|ω₁}} there is at least one infinite {{mvar|α}} with {{mvar|A ∩ α {{=}} A_α}}.

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}}} is stationary in {{math|κ}}

The principle {{math|◊_ω1}} 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|ω₁}} there is a closed unbounded subset {{math|C}} of {{math|ω₁}} such that for all {{math|α}} in {{math|C}} we have {{math|A ∩ α ∈ A_α}} and {{math|C ∩ α ∈ A_α}}.

{{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 $\backslash diamondsuit\_\backslash kappa$ equivalent to a property of partitions of $\backslash kappa$ with diagonal intersection of initial segments of the partitions stationary in $\backslash kappa$.P. Matet, "[https://eudml.org/doc/211737 On diamond sequences]". Fundamenta Mathematicae vol. 131, iss. 1, pp.35--44 (1988)

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 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.

• List of statements independent of ZFC
• Statements true in L

{{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}}

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Category:Set theory

Category:Mathematical principles

Category:Independence results

Category:Constructible universe