鑽石原則（）是由在引入的組合原理，它適用於（）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（）意味著存在。
定義
The diamond principle says that there exists a **', in other words sets for such that for any subset of ω1 the set of with is stationary in .
There are several equivalent forms of the diamond principle. One states that there is a countable collection of subsets of for each countable ordinal such that for any subset of there is a stationary subset of such that for all in we have and . Another equivalent form states that there exist sets for such that for any subset of there is at least one infinite with .
More generally, for a given cardinal number and a stationary set , the statement (sometimes written or ) is the statement that there is a sequence such that
- each
- for every , {{math|{α ∈ S : A ∩ α Aα}}} is stationary in
The principle is the same as .
The diamond-plus principle states that there exists a -sequence, in other words a countable collection of subsets of for each countable ordinal α such that for any subset of there is a closed unbounded subset of such that for all in we have and .
屬性和使用
showed that the diamond principle implies the existence of Suslin trees. He also showed that 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 implies , but Shelah gave models of , 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.
used to construct a -algebra serving as a counterexample to Naimark's problem.
For all cardinals and stationary subsets , holds in the constructible universe. proved that for , follows from for stationary that do not contain ordinals of cofinality .
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
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- ZFC系統無法確定的命題列表
參考文獻
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