Reflecting cardinal

Reflecting cardinal

In set theory, a mathematical discipline, a reflecting cardinal is a cardinal number &kappa; for which there is a normal ideal "I" on &kappa; such that for every "X"&isin;"I"+, the set of α&isin;&kappa; for which "X" reflects at α is in "I"+. (A stationary subset "S" of &kappa; is said to reflect at α<&kappa; if "S"&cap;α is stationary in α.)Reflecting cardinals were introduced by harv|Mekler|Shelah|1989.

Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals.Every reflecting cardinal is a greatly Mahlo cardinal, and is also a limit of greatly Mahlo cardinals, where a cardinal &kappa; is called greatly Mahlo if it is &kappa;+-Mahlo.

References

*Citation | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | publisher=Springer-Verlag | location=Berlin, New York | edition=third millennium | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003|page=697
*Citation | last1=Mekler | first1=Alan H. | last2=Shelah | first2=Saharon | author2-link=Saharon Shelah | title=The consistency strength of ``every stationary set reflects" | id=MathSciNet | id = 1029909 | year=1989 | journal=Israel Journal of Mathematics | issn=0021-2172 | volume=67 | issue=3 | pages=353–366


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