In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, a mathematical discipline, a reflecting cardinal is a
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
κ for which there is a normal
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
''I'' on κ such that for every ''X''∈''I''
+, the set of α∈κ for which ''X'' reflects at α is in ''I''
+. (A
stationary subset In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three clo ...
''S'' of κ is said to reflect at α<κ if ''S''∩α is stationary in α.)
Reflecting cardinals were introduced by .
Every
weakly compact cardinal In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally ...
is a reflecting cardinal, and is also a limit of reflecting cardinals.
The consistency strength of an inaccessible reflecting cardinal is strictly greater than a greatly Mahlo cardinal, where a cardinal κ is called greatly Mahlo if it is
κ+-Mahlo . An inaccessible reflecting cardinal is not in general Mahlo however, see https://mathoverflow.net/q/212597.
See also
*
List of large cardinal properties
This page includes a list of cardinals with large cardinal properties. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a g ...
References
*
*
Large cardinals
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