钻石原则)是由罗纳德·詹森英语Ronald JensenJensen (1972)引入的组合原理,它适用于哥德尔可构造全集英语Gödel's constructible universeL)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公理英语Axiom of constructibilityV = L)意味着存在苏斯林树英语Suslin tree


定义

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The diamond principle says that there exists a ◊-sequence, in other words sets Aαα for α < ω1 such that for any subset A of ω1 the set of α with Aα = 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 AαAα and CαAα. Another equivalent form states that there exist sets Aαα for α < ω1 such that for any subset A of ω1 there is at least one infinite α with Aα = Aα.

More generally, for a given cardinal number κ and a stationary set Sκ, the statement S (sometimes written ◊(S) or κ(S)) is the statement that there is a sequence Aα : αS such that

  • each Aαα
  • for every Aκ, {αS : Aα = 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 AαAα and CαAα.

属性和使用

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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 Sκ+, 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

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参考文献

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