鑽石原則

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 ,  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

 ZFC系統無法確定的命題列表

 參考文獻 

Category:集合论
Category:数学原理
Category:獨立結果
Category:可构造宇宙