**鑽石原則**（◊）是由羅納德·詹森（英语：）在Jensen (1972)引入的組合原理，它適用於哥德尔可构造全集（英语：）（*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（英语：）（*V* = *L*）意味著存在蘇斯林樹（英语：）。

定義
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已隱藏部分未翻譯内容，歡迎參與翻譯。
The diamond principle ◊ says that there exists a **◊-sequence**, in other words sets *Aα* ⊆ *α* for *α* < *ω*1 such that for any subset *A* of ω1 the set of *α* with *A* ∩ *α* = *Aα* is stationary in *ω*1.

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 *ω*1 there is a stationary subset *C* of *ω*1 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* < *ω*1 such that for any subset *A* of *ω*1 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 ⟨*Aα* : *α* ∈ *S*⟩ such that

* each *Aα* ⊆ *α*
* for every *A* ⊆ *κ*, {*α* ∈ *S* : *A* ∩ *α* = *Aα*} is stationary in *κ*

The principle ◊*ω*1 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 *ω*1 there is a closed unbounded subset *C* of *ω*1 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
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* 哥德爾定理
* 哥德爾完備性定理
* ZFC公理體系（英语：）
* ZFC系統無法確定的命題列表

參考文獻
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* Akemann, Charles; Weaver, Nik.. Proceedings of the National Academy of Sciences. 2004, **101** (20): 7522–7525. Bibcode:2004PNAS..101.7522A. MR 2057719. PMC 419638 可免费查阅. PMID 15131270. arXiv:math.OA/0312135 可免费查阅. doi:10.1073/pnas.0401489101 可免费查阅.
* Jensen, R. Björn.. Annals of Mathematical Logic. 1972, **4** (3): 229–308. MR 0309729. doi:10.1016/0003-4843(72)90001-0 可免费查阅.
* Rinot, Assaf.. . Contemporary Mathematics **533**. Providence, RI: AMS. 2011: 125–156. Bibcode:2009arXiv0911.2151R. ISBN 978-0-8218-4812-8. MR 2777747. arXiv:0911.2151 可免费查阅.
* Shelah, Saharon.. Israel Journal of Mathematics（英语：）. 1974, **18** (3): 243–256. MR 0357114. S2CID 123351674. doi:10.1007/BF02757281 可免费查阅.
* Shelah, Saharon.. Proceedings of the American Mathematical Society（英语：）. 2010, **138** (6): 2151–2161. doi:10.1090/S0002-9939-10-10254-8 可免费查阅.