Diamond principle

(Diamonds) is a " combinatorial " principle in set theory.

Definition

For every infinite cardinal number is an abbreviation for the following statement:

• There is a sequence with the following properties:

• Applies to all
• For all is the amount a stationary subset of.

One often speaks of simplicity assume that the principle makes it possible to "guess" subsets of to. While the number of subsets of (ie, the cardinality of the power set of ) is greater than for the set of Cantor indeed, postulated that there is a transfinite sequence of length, all subsets of "guesses " (more precisely, stationary often better and better approximated ).

Instead you write often.

Associated with CH and GCH

◊ The statement is in the Zermelo -Fraenkel set theory ( ZFC ) neither provable nor refutable.

It is easy to that of the continuum hypothesis CH ◊ follows. More generally follows from the equation. From CH ◊ one can not conclude, however, from along with you can close. From the generalized continuum hypothesis GCH therefore follows for all with uncountable cofinality.

Applications

◊ implies that the Suslin hypothesis is false; in other words, that there is a Suslin line, ie a non- separable linear order in which yet every family of disjoint intervals is at most countable.

• Set theory