鑽石原則 - Wikiwand

定义

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The diamond principle $◊$ says that there exists a ◊-sequence, in other words sets $A α \subseteq α$ for $α < ω 1$ such that for any subset $A$ of ω₁ the set of $α$ with $A \cap α = A α$ 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 $A \cap α \in A α$ and $C \cap α \in A α$ . Another equivalent form states that there exist sets $A α \subseteq α$ for $α < ω 1$ such that for any subset $A$ of $ω 1$ there is at least one infinite $α$ with $A \cap α = A α$ .

More generally, for a given cardinal number $κ$ and a stationary set $S \subseteq κ$ , the statement $◊ S$ (sometimes written $◊(S)$ or $◊ κ (S)$ ) is the statement that there is a sequence $⟨ A α : α \in S ⟩$ such that

each $A α \subseteq α$
for every $A \subseteq κ$ , ${α \in S : A \cap α = A α}$ 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 $A \cap α \in A α$ and $C \cap α \in A α$ .

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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 $♣ + \neg CH$ , 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 $S \subseteq κ +$ , $◊ S$ holds in the constructible universe. Shelah (2010) proved that for $κ > ℵ 0$ , $◊ κ + (S)$ 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.

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钻石原则

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属性和使用

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