Ineffable cardinal explained

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by . In the following definitions,

\kappa will always be a regular uncountable cardinal number.\kappa is called almost ineffable if for every f: \kappa \to l{P}(\kappa) (where l{P}(\kappa) is the powerset of \kappa ) with the property that f(\delta) is a subset of \delta for all ordinals \delta < \kappa , there is a subset S of \kappa having cardinality \kappa and homogeneous for f , in the sense that for any \delta _{1} < \delta _{2} in S , f(\delta _{1)} = f(\delta _{2)} \cap \delta _{1} .\kappa is called ineffable if for every binary-valued function f : [\kappa] ^{2\to} \{ 0 , 1 \} , there is a stationary subset of \kappa on which f is homogeneous: that is, either f maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one. An equivalent formulation is that a cardinal \kappa is ineffable if for every sequence such that each, there is such that