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), A_δ ⊂ δ there are α, β, belonging to C, with α<β, such that A_α=A_β∩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

Harvey Friedman: "Subtle Cardinals and Linear Orderings." Annals of Pure and Applied Logic (January 15, 2001) 107(1-3):1-34.