Subtle cardinal
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In mathematics, a subtle cardinal is a certain kind of large cardinal number.
Formally, a cardinal κ is subtle iff for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ (for an arbitrary δ), Aδ ⊂ δ there are α, β belonging to C such that Aα=Aβ∩α.
Theorem
There is a subtle cardinal ≤κ iff every transitive set S of cardinality κ contains x and y such that x is a proper subset of y and x ≠ Ø and x ≠ {Ø}. An infinite ordinal κ is subtle iff for every λ<κ, every transitive set S of cardinality κ includes a chain (under inclusion) of order type λ.Template:Math-stub