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In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by Ronald Jensen in that holds in the constructible universe and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the Axiom of constructibility (V=L) implies the existence of a Suslin tree. == Definitions == The diamond principle ◊ says that there exists a ◊-sequence, in other words sets ''A''α⊆α for α<ω1 such that for any subset ''A'' of ω1 the set of α with ''A''∩α = ''A''α 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 ''A''∩α ∈ Aα and ''C''∩α ∈ Aα. Another equivalent form states that there exist sets ''A''α⊆α for α<ω1 such that for any subset ''A'' of ω1 there is at least one infinite α with ''A''∩α = ''A''α. More generally, for a given cardinal number and a stationary set , the statement ◊''S'' (sometimes written ◊(''S'') or ◊κ(''S'')) is the statement that there is a sequence such that * each * for every is stationary in The principle ◊ω1 is the same as ◊. The diamond plus principle ◊+ says 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 ''A''∩α ∈ Aα and ''C''∩α ∈ Aα. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Diamond principle」の詳細全文を読む スポンサード リンク
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