Subtle cardinal

Subtle cardinal

In mathematics, a subtle cardinal is a certain kind of large cardinal number.

Formally, a cardinal &kappa; is subtle if and only if for every closed and unbounded "C" &sub; &kappa; and for every sequence "A" of length &kappa; for which element number &delta; (for an arbitrary &delta;), A&delta; &sub; &delta; there are α, &beta;, belonging to "C", with α<&beta;, such that "A"α="A"&beta;&cap;α.

Theorem

There is a subtle cardinal &le;&kappa; if and only if every transitive set "S" of cardinality &kappa; contains "x" and "y" such that "x" is a proper subset of "y" and "x" &ne; &Oslash; and "x" &ne; {&Oslash;}. An infinite ordinal &kappa; is subtle if and only if for every &lambda;<&kappa;, every transitive set "S" of cardinality &kappa; includes a chain (under inclusion) of order type &lambda;.

References

*citation|first=Harvey |last=Friedman|authorlink=Harvey Friedman|title=Subtle Cardinals and Linear Orderings|journal= Annals of Pure and Applied Logic |year= 2001|volume=107|issue=1-3|pages=1-34
doi=10.1016/S0168-0072(00)00019-1


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