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 if and only if 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, with α<β, such that Aα=Aβ∩α.
[edit] Theorem
There is a subtle cardinal ≤κ if and only if 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 if and only if for every λ<κ, every transitive set S of cardinality κ includes a chain (under inclusion) of order type λ.
[edit] References
- Friedman, Harvey (2001), “Subtle Cardinals and Linear Orderings”, Annals of Pure and Applied Logic 107 (1-3): 1-34, DOI 10.1016/S0168-0072(00)00019-1
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