PlanetMath: $\Diamond$

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$\Diamond$

(Definition)

Definition 1 Let $S\subseteq\kappa$ be a stationary set. Then the combinatorial principle $\Diamond_S$ holds if and only if there is a sequence $\langle A_{\alpha}\rangle_{\alpha\in S}$ such that each $A_{\alpha}\subseteq\alpha$ and for any $A\subseteq\kappa$ , $\{\alpha\in S\mid A\cap\alpha=A_{\alpha}\}$ is stationary.

To get some sense of what this means, observe that for any $\lambda<\kappa$ , $\{\lambda\}\subseteq\kappa$ , so the set of $A_\alpha=\{\lambda\}$ is stationary (in $\kappa$ ). More strongly, suppose $\kappa>\lambda$ . Then any subset of $T\subset\lambda$ is bounded in $\kappa$ so $A_\alpha=T$ on a stationary set. Since $\vert S\vert=\kappa$ , it follows that $2^\lambda\leq\kappa$ . Hence $\Diamond_{\aleph_1}$ , the most common form (often written as just $\Diamond$ ), implies CH.

C. Akemann and N. Weaver used $\Diamond$ to construct a -algebra serving as a counterexample to Naimark's problem.

Bibliography

1: Akemann, C., and N. Weaver, Consistency of a counterexample to Naimark's problem. Preprint available on the arXiv at http://arxiv.org/abs/math.OA/0312135.

" $\Diamond$ " is owned by Henry. [ full author list (2) | owner history (2) ]

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See Also: $\clubsuit$ , $\Diamond$ is equivalent to $\clubsuit$ and continuum hypothesis, proof of $\Diamond$ is equivalent to $\clubsuit$ and continuum hypothesis, combinatorial principle

Other names:

diamond

Also defines:

$\Diamond_S$

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Cross-references: counterexample, CH, implies, bounded, subset, stationary, sequence, combinatorial principle, stationary set
There are 5 references to this entry.

This is version 5 of $\Diamond$ , born on 2002-07-31, modified 2004-04-10.
Object id is 3245, canonical name is Diamond.
Accessed 4281 times total.

Classification:

AMS MSC:

03E65 (Mathematical logic and foundations :: Set theory :: Other hypotheses and axioms)

Pending Errata and Addenda

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