Difference between revisions of "Diamond principle"
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Results: | 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 $κ$).<cite>Ketonen1974:SomeCombinatorialPrinciples</cite> | * 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 $κ$).<cite>Ketonen1974:SomeCombinatorialPrinciples</cite> | ||
* If $κ$ is [[ineffable]], weakly ineffable or [[subtle]], then $♢_κ$ holds.<cite>JensenKunen1969:Ineffable</cite><cite>Ketonen1974:SomeCombinatorialPrinciples</cite> | * If $κ$ is [[ineffable]], weakly ineffable or [[subtle]], then $♢_κ$ holds.<cite>JensenKunen1969:Ineffable</cite><cite>Ketonen1974:SomeCombinatorialPrinciples</cite> |
Latest revision as of 12:47, 6 October 2019
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
- Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www bibtex
- Ketonen, Jussi. Some combinatorial principles. Trans Amer Math Soc 188:387-394, 1974. DOI bibtex
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