Limit cardinal

In mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.

A cardinal λ is a strong limit cardinal if λ cannot be reached by repeated powerset operations. This means that λ is nonzero and, for all κ < λ, 2^κ < λ. Every strong limit cardinal is also a weak limit cardinal, because κ⁺ ≤ 2^κ for every cardinal κ, where κ⁺ denotes the successor cardinal of κ.

The first infinite cardinal, ${\displaystyle \aleph _{0}}$ (aleph-naught), is a strong limit cardinal, and hence also a weak limit cardinal.

One way to construct limit cardinals is via the union operation: ${\displaystyle \aleph _{\omega }}$ is a weak limit cardinal, defined as the union of all the alephs before it; and in general ${\displaystyle \aleph _{\lambda }}$ for any limit ordinal λ is a weak limit cardinal.

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