Diamond principle

Diamond principle $♢$ (TeX \diamondsuit) is the statement what?...

Variants:

$♢_κ$ or $♢(κ)$ is the statement “There exist sets $(S_α|α < κ)$, $S_α ⊆ α$ ($α < κ$) such that for any $X ∈ κ$ the set $\{α|X ∩ α = S_α\}$ is Mahlo” introduced in [1].[2]

Results:

• $♢$ holds in the constructible universe $L$.
• If $κ$ is ethereal and $2^\underset{\smile}{κ} = κ$, then $♢_κ$ holds (where $2^\underset{\smile}{κ} = \bigcup \{ 2^α | α < κ \}$ is the weak power of $κ$).[2]
• If $κ$ is ineffable, weakly ineffable or subtle, then $♢_κ$ holds.[1][2]

References

1. Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www   bibtex
2. Ketonen, Jussi. Some combinatorial principles. Trans Amer Math Soc 188:387-394, 1974. DOI   bibtex

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