钻石原则
钻石原则(◊)是由罗纳德·詹森在Jensen (1972)引入的组合原理,它适用于哥德尔可构造全集(L)并暗示了连续统假设。罗纳德·詹森在证明中提取了钻石原理,即constructibility公理(V = L)意味着存在苏斯林树。
定义
The diamond principle ◊ says that there exists a ◊-sequence, in other words sets Aα ⊆ α for α < ω1 such that for any subset A of ω1 the set of α with A ∩ α = Aα is stationary in ω1.
There are several equivalent forms of the diamond principle. One states that there is a countable collection Aα of subsets of α for each countable ordinal α such that for any subset A of ω1 there is a stationary subset C of ω1 such that for all α in C we have A ∩ α ∈ Aα and C ∩ α ∈ Aα. Another equivalent form states that there exist sets Aα ⊆ α for α < ω1 such that for any subset A of ω1 there is at least one infinite α with A ∩ α = Aα.
More generally, for a given cardinal number κ and a stationary set S ⊆ κ, the statement ◊S (sometimes written ◊(S) or ◊κ(S)) is the statement that there is a sequence ⟨Aα : α ∈ S⟩ such that
- each Aα ⊆ α
- for every A ⊆ κ, {α ∈ S : A ∩ α = Aα} is stationary in κ
The principle ◊ω1 is the same as ◊.
The diamond-plus principle ◊+ states that there exists a ◊+-sequence, in other words a countable collection Aα of subsets of α for each countable ordinal α such that for any subset A of ω1 there is a closed unbounded subset C of ω1 such that for all α in C we have A ∩ α ∈ Aα and C ∩ α ∈ Aα.
属性和使用
Jensen (1972) showed that the diamond principle ◊ implies the existence of Suslin trees. He also showed that 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 ♣ + CH implies ◊, but Shelah gave models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).
The diamond principle ◊ does not imply the existence of a Kurepa tree, but the stronger ◊+ principle implies both the ◊ principle and the existence of a Kurepa tree.
Akemann & Weaver (2004) used ◊ to construct a C*-algebra serving as a counterexample to Naimark's problem.
For all cardinals κ and stationary subsets S ⊆ κ+, ◊S holds in the constructible universe. Shelah (2010) proved that for κ > ℵ0, ◊κ+(S) follows from 2κ = κ+ for stationary S that do not contain ordinals of cofinality κ.
Shelah showed that the diamond principle solves the Whitehead problem by implying that every Whitehead group is free.
参见
- 哥德尔定理
- 哥德尔完备性定理
- ZFC公理体系
- ZFC系统无法确定的命题列表
参考文献
- Akemann, Charles; Weaver, Nik. Consistency of a counterexample to Naimark's problem. Proceedings of the National Academy of Sciences. 2004, 101 (20): 7522–7525. Bibcode:2004PNAS..101.7522A. MR 2057719. PMC 419638 . PMID 15131270. arXiv:math.OA/0312135 . doi:10.1073/pnas.0401489101 .
- Jensen, R. Björn. The fine structure of the constructible hierarchy. Annals of Mathematical Logic. 1972, 4 (3): 229–308. MR 0309729. doi:10.1016/0003-4843(72)90001-0 .
- Rinot, Assaf. Jensen's diamond principle and its relatives. Set theory and its applications. Contemporary Mathematics 533. Providence, RI: AMS. 2011: 125–156. Bibcode:2009arXiv0911.2151R. ISBN 978-0-8218-4812-8. MR 2777747. arXiv:0911.2151 .
- Shelah, Saharon. Infinite Abelian groups, Whitehead problem and some constructions. Israel Journal of Mathematics. 1974, 18 (3): 243–256. MR 0357114. S2CID 123351674. doi:10.1007/BF02757281 .
- Shelah, Saharon. Diamonds. Proceedings of the American Mathematical Society. 2010, 138 (6): 2151–2161. doi:10.1090/S0002-9939-10-10254-8 .