# $\clubsuit$

$\clubsuit_{S}$ is a combinatoric principle weaker than $\Diamond_{S}$. It states that, for $S$ stationary in $\kappa$, there is a sequence $\langle A_{\alpha}\rangle_{\alpha\in S}$ such that $A_{\alpha}\subseteq\alpha$ and $\operatorname{sup}(A_{\alpha})=\alpha$ and with the property that for each unbounded subset $T\subseteq\kappa$ there is some $A_{\alpha}\subseteq X$.

Any sequence satisfying $\Diamond_{S}$ can be adjusted so that $\operatorname{sup}(A_{\alpha})=\alpha$, so this is indeed a weakened form of $\Diamond_{S}$.

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 $\operatorname{sup}(A_{\alpha})=\alpha$, $\alpha$ is also the limit of a subsequence of $C^{*}$ and therefore an element of $C$.

Title $\clubsuit$ clubsuit 2013-03-22 12:53:52 2013-03-22 12:53:52 Henry (455) Henry (455) 4 Henry (455) Definition msc 03E65 clubsuit Diamond DiamondIsEquivalentToClubsuitAndContinuumHypothesis ProofOfDiamondIsEquivalentToClubsuitAndContinuumHypothesis CombinatorialPrinciple