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 (Jensen & Kunen, 1969; Ketonen, 1974)

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 $κ$).(Ketonen, 1974)

• If $κ$ is ineffable, weakly ineffable or subtle, then $♢_κ$ holds.(Jensen & Kunen, 1969; Ketonen, 1974)

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References

1. Jensen, R., & Kunen, K. (1969). Some combinatorial properties of L and V. http://www.mathematik.hu-berlin.de/ raesch/org/jensen.html
2. Ketonen, J. (1974). Some combinatorial principles. Trans. Amer. Math. Soc., 188, 387–394. https://doi.org/10.1090/S0002-9947-1974-0332481-5

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