| 词条 | Diamond principle |
| 释义 |
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. DefinitionsThe diamond principle {{math|◊}} says that there exists a {{vanchor|◊-sequence}}, in other words sets {{math|Aα ⊆ α}} for {{math|α < ω1}} such that for any subset {{math|A}} of ω1 the set of {{math|α}} with {{math|A ∩ α {{=}} Aα}} is stationary in {{math|ω1}}. There are several equivalent forms of the diamond principle. One states that there is a countable collection {{math|Aα}} 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 ∈ Aα}} and {{math|C ∩ α ∈ Aα}}. Another equivalent form states that there exist sets {{math|Aα ⊆ 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 {{=}} Aα}}. 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|⟨Aα : α ∈ S⟩}} such that
The principle {{math|◊ω1}} is the same as {{math|◊}}. The diamond-plus principle {{math|◊+}} says that there exists a {{math|◊+}}-sequence, in other words a countable collection {{math|Aα}} 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 ∈ Aα}} and {{math|C ∩ α ∈ Aα}}. 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|◊}}).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|2κ {{=}} 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
References{{refbegin|30em}}
|MR=2777747|title= Set theory and its applications|pages= 125–156|series=Contemporary Mathematics|volume= 533|publisher= AMS|place= Providence, RI|year= 2011|isbn= 978-0-8218-4812-8|arxiv=0911.2151|bibcode=2009arXiv0911.2151R}}
|journal=Israel Journal of Mathematics |volume=18 |year=1974|pages=243–256|doi=10.1007/BF02757281| mr=0357114| issue=3}}
3 : Set theory|Mathematical principles|Independence results |
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