Subtle cardinal explained

In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.

A cardinal

\kappa is called subtle if for every closed and unbounded C\subset\kappa and for every sequence A of length \kappa such that A _{\delta\subset\delta} for arbitrary \delta<\kappa (where A _{\delta} is the \delta th element), there exist \alpha,\beta , belonging to C , with \alpha<\beta , such that A _{\alpha=A} _{\beta\cap\alpha} .

A cardinal

\kappa is called ethereal if for every closed and unbounded C\subset\kappa and for every sequence A of length \kappa such that A _{\delta\subset\delta} and A _{\delta} has the same cardinality as \delta for arbitrary \delta<\kappa , there exist \alpha,\beta , belonging to C , with \alpha<\beta , such that rm{card}(\alpha)= card (A _{\beta\cup} A _{\alpha)} .

Subtle cardinals were introduced by . Ethereal cardinals were introduced by . Any subtle cardinal is ethereal,^{p. 388} and any strongly inaccessible ethereal cardinal is subtle.^{p. 391}

Characterizations

Some equivalent properties to subtlety are known.

Relationship to Vopěnka's Principle

\kappa is subtle if and only if in V _{\kappa+1} , any logic has stationarily many weak compactness cardinals.^[1]

Vopenka's principle itself may be stated as the existence of a strong compactness cardinal for each logic.

Chains in transitive sets

There is a subtle cardinal

\leq\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 ≠ \varnothing and x ≠ \{\varnothing\} .^{[2] Corollary 2.6} An infinite ordinal \kappa is subtle if and only if for every λ <\kappa , every transitive set S of cardinality \kappa includes a chain (under inclusion) of order type λ .

Extensions

A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.^{[3] p.1014}

See also

• List of large cardinal properties

References

Citations

Notes and References

1. W. Boney, S. Dimopoulos, V. Gitman, M. Magidor "Model Theoretic Characterizations of Large Cardinals Revisited"
2. [Harvey Friedman|H. Friedman]
3. C. Henrion, "Properties of Subtle Cardinals. Journal of Symbolic Logic, vol. 52, no. 4 (1987), pp.1005--1019."