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. A cardinal number \kappa is called almost ineffable if for every (where is the powerset of \kappa) with the property that f(\delta) is a subset of \delta for all ordinals, there is a subset S of \kappa having cardinality \kappa and homogeneous for f, in the sense that for any in S,. A cardinal number \kappa is called ineffable if for every binary-valued function, 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 is stationary in κ . Another equivalent formulation is that a regular uncountable cardinal \kappa is ineffable if for every set S of cardinality \kappa of subsets of \kappa, there is a normal (i.e. closed under diagonal intersection) non-trivial \kappa-complete filter \mathcal F on \kappa deciding S: that is, for any X\in S, either or. This is similar to a characterization of weakly compact cardinals. More generally, \kappa is called n-ineffable (for a positive integer n) if for every there is a stationary subset of \kappa on which f is n-homogeneous (takes the same value for all unordered n-tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable. Ineffability is strictly weaker than 3-ineffability. p. 399 A totally ineffable cardinal is a cardinal that is n-ineffable for every. If \kappa is (n+1)-ineffable, then the set of n-ineffable cardinals below \kappa is a stationary subset of \kappa. Every n-ineffable cardinal is n-almost ineffable (with set of n-almost ineffable below it stationary), and every n-almost ineffable is n-subtle (with set of n-subtle below it stationary). The least n-subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least n-almost ineffable is \Pi^1_2-describable), but (n-1)-ineffable cardinals are stationary below every n-subtle cardinal. A cardinal κ is completely ineffable if there is a non-empty such that

• every A \in R is stationary
• for every A \in R and, there is homogeneous for f with B \in R. Using any finite n > 1 in place of 2 would lead to the same definition, so completely ineffable cardinals are totally ineffable (and have greater consistency strength). Completely ineffable cardinals are \Pi^1_n-indescribable for every n, but the property of being completely ineffable is \Delta^2_1. The consistency strength of completely ineffable is below that of 1-iterable cardinals, which in turn is below remarkable cardinals, which in turn is below ω-Erdős cardinals. A list of large cardinal axioms by consistency strength is available in the section below.

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