Diamond principle

In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe (L) 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.

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 A_α 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 ∈ A_α and C∩α ∈ A_α. 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 ${\displaystyle \kappa }$ and a stationary set ${\displaystyle S\subseteq \kappa }$, the statement ◊_S (sometimes written ◊(S) or ◊_κ(S)) is the statement that there is a sequence ${\displaystyle \langle A_{\alpha }:\alpha \in S\rangle }$ such that

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