鑽石原則

鑽石原則（◊）是由羅納德·詹森（英語：Ronald Jensen）在Jensen (1972)引入的組合原理，它適用於哥德爾可構造全集（英語：Gödel's constructible universe）（L）並暗示了連續統假設。羅納德·詹森在證明中提取了鑽石原理，即constructibility公理（英語：Axiom of constructibility）（V = L）意味着存在蘇斯林樹（英語：Suslin tree）。

定義

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

There are several equivalent forms of the diamond principle. One states that there is a countable collection A_α 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 ∈ A_α and C ∩ α ∈ A_α. Another equivalent form states that there exist sets A_α ⊆ α for α < ω₁ such that for any subset A of ω₁ there is at least one infinite α with A ∩ α = A_α.

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 ⟨A_α : α ∈ S⟩ such that

• each A_α ⊆ α
• for every A ⊆ κ, {α ∈ S : A ∩ α = A_α} 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 A_α 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 ∈ A_α and C ∩ α ∈ A_α.

屬性和使用

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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 2^κ = 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.

參見

• 哥德爾定理
• 哥德爾完備性定理
• ZFC公理體系（英語：Zermelo-Fraenkel Set Theory）
• ZFC系統無法確定的命題列表

參考文獻

• 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（英語：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（英語：Proceedings of the American Mathematical Society）. 2010, 138 (6): 2151–2161. doi:10.1090/S0002-9939-10-10254-8 .