NationMaster - Encyclopedia: Diamondsuit

In mathematics, and particularly in axiomatic set theory, $Diamond_kappa (S)$ (diamondsuit or diamond) is a certain family of combinatorial principles. Euclid, detail from The School of Athens by Raphael. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... In proving results in combinatorics several useful combinatorial rules or combinatorial principles are used. ...

1 Definition
2 Properties and use
3 References
4 See also

Definition

For a given cardinal number $κ$ and a stationary set $Ssubseteqkappa$ , $Diamond_kappa (S)$ is the statement that there is a sequence $langle A_alpha: alpha in S rangle$ such that In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... In mathematics, particularly in set theory, if is a cardinal, , and intersects every club in , then is called a stationary set. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

each $A_alpha subseteq alpha$
for every $A subseteq kappa, {alpha in S: A cap alpha = A_alpha}$ is stationary in $κ$

When $Diamond_kappa (S)$ is written $Diamond_kappa$ , and $Diamond_{omega_1}$ is written $Diamond$

Properties and use

It can be shown that ◊ ⇒ CH; also, ♣ + CH ⇒ ◊, but there also exist models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊). In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a combinatorial principle that is a weaker version of ◊S and that was introduced in 1975 by A. Ostaszewski. ...

Charles Akemann and Nik Weaver used ◊ to construct a C^*-algebra serving as a counterexample to Naimark's problem. C*-algebras are an important area of research in functional analysis, a branch of mathematics. ... In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ... In mathematics, Naimarks problem is a question in functional analysis. ...

For all cardinals $κ$ and stationary subsets $S subseteq kappa^+$ , $Diamond_{kappa^+} (S)$ holds in the constructible universe. In mathematics, the constructible universe (or GÃ¶dels constructible universe), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. ...

References

Charles Akemann, Nik Weaver, Consistency of a counterexample to Naimark's problem, online

Contents

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Properties and use

References

See also

Definition