# $\mathrm{\u2663}$

${\mathrm{\u2663}}_{S}$ is a combinatoric principle weaker than ${\mathrm{\u25c7}}_{S}$. It states that, for $S$ stationary in $\kappa $, there is a sequence ${\u27e8{A}_{\alpha}\u27e9}_{\alpha \in S}$ such that ${A}_{\alpha}\subseteq \alpha $ and $\mathrm{sup}({A}_{\alpha})=\alpha $ and with the property that for each unbounded^{} subset $T\subseteq \kappa $ there is some ${A}_{\alpha}\subseteq X$.

Any sequence satisfying ${\mathrm{\u25c7}}_{S}$ can be adjusted so that $\mathrm{sup}({A}_{\alpha})=\alpha $, so this is indeed a weakened form of ${\mathrm{\u25c7}}_{S}$.

Any such sequence actually contains a stationary set of $\alpha $ such that ${A}_{\alpha}\subseteq T$ for each $T$: given any club $C$ and any unbounded $T$, construct a $\kappa $ sequence, ${C}^{*}$ and ${T}^{*}$, from the elements of each, such that the $\alpha $-th member of ${C}^{*}$ is greater than the $\alpha $-th member of ${T}^{*}$, which is in turn greater than any earlier member of ${C}^{*}$. Since both sets are unbounded, this construction is possible, and ${T}^{*}$ is a subset of $T$ still unbounded in $\kappa $. So there is some $\alpha $ such that ${A}_{\alpha}\subseteq {T}^{*}$, and since $\mathrm{sup}({A}_{\alpha})=\alpha $, $\alpha $ is also the limit of a subsequence of ${C}^{*}$ and therefore an element of $C$.

Title | $\mathrm{\u2663}$ |
---|---|

Canonical name | clubsuit |

Date of creation | 2013-03-22 12:53:52 |

Last modified on | 2013-03-22 12:53:52 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 4 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03E65 |

Synonym | clubsuit |

Related topic | Diamond^{} |

Related topic | DiamondIsEquivalentToClubsuitAndContinuumHypothesis |

Related topic | ProofOfDiamondIsEquivalentToClubsuitAndContinuumHypothesis |

Related topic | CombinatorialPrinciple |