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Limit Cardinal

A nonzero cardinal number is a limit cardinal if it is not the least cardinal number greater than some other cardinal number. Equivalently, it is the supremum of smaller cardinal numbers. Assuming the axiom of choice, the limit cardinals are

aleph_0 and aleph_lambda,

for nonzero limit ordinals lambda. Thus aleph_0 is a limit of the finite cardinal numbers, and the first uncountable limit cardinal is aleph_omega.

Some authors reserve the phrase "limit cardinal" for uncountable cardinals, in which convention aleph_omega is the first example. A limit cardinal need not be a strong limit cardinal, meaning that it need not satisfy 2^lambda<kappa for every lambda<kappa.

See also

Aleph, Cardinal Number, Limit Ordinal, Strong Limit Cardinal

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References

Jech, T. Set Theory, 3rd millennium ed. Berlin, Germany: Springer-Verlag, 2003.

Cite this as:

Weisstein, Eric W. "Limit Cardinal." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LimitCardinal.html

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