Diamonds (set theory)

${\displaystyle \Diamond }$( Diamond ) is a "combinatorial" principle in set theory .

definition

For each infinite cardinal number is an abbreviation for the following statement: ${\displaystyle \kappa }$ ${\displaystyle \Diamond _{\kappa }}$

• there is a sequence with the following properties: ${\displaystyle \langle A_{\alpha }:\alpha \in \kappa \rangle }$

• for all true ${\displaystyle \alpha }$ ${\displaystyle A_{\alpha }\subseteq \alpha }$
• for all the set is a stationary subset of . ${\displaystyle A\subseteq \kappa }$ ${\displaystyle \{\alpha \in \kappa :A\cap \alpha =A_{\alpha }\}}$ ${\displaystyle \kappa }$

Often one speaks in a simplistic way that the principle makes it possible to "guess" subsets of . While the number of subsets of (i.e. the cardinality of the power set of ) is greater than according to Cantor's theorem , postulates that there is a transfinite sequence of length that “guesses” all subsets of (more precisely: stationary often better and better approximated). ${\displaystyle \Diamond _{\kappa }}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \Diamond _{\kappa }}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$

Instead of just writing . ${\displaystyle \Diamond _{\omega _{1}}}$ ${\displaystyle \Diamond }$

Connection with CH and GCH

The statement ◊ in the Zermelo-Fraenkel set theory (ZFC) can neither be proven nor refuted.

It is easy to show that ◊ implies the continuum hypothesis CH. More generally, the equation follows . From CH can ◊ not conclude, but from with one can close. So it follows from the generalized continuum hypothesis GCH for all with uncountable cofinality . ${\displaystyle \Diamond _{\kappa ^{+}}}$ ${\displaystyle 2^{\kappa }=\kappa ^{+}}$ ${\displaystyle 2^{\kappa }=\kappa ^{+}}$ ${\displaystyle \kappa ^{\omega }=\kappa }$ ${\displaystyle \Diamond _{\kappa ^{+}}}$ ${\displaystyle \Diamond _{\kappa ^{+}}}$ ${\displaystyle \kappa }$

Applications

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