Aleph-0

The Set Theory symbol for a Set having the same Cardinal Number as the ``small'' Infinite Set of Integers. The Algebraic Numbers also belong to $\aleph_0$. Rather surprising properties satisfied by $\aleph_0$ include

$${\aleph_0}^r=\aleph_0$$ (1)

$$r{\aleph_0}=\aleph_0$$ (2)

$${\aleph_0}+f=\aleph_0,$$ (3)

where $f$ is any Finite Set. However,

$${\aleph_0}^{\aleph_0}=C,$$ (4)

where $C$ is the Continuum.

See also Aleph-1, Cardinal Number, Continuum, Continuum Hypothesis, Countably Infinite Set, Finite, Infinite, Transfinite Number, Uncountably Infinite Set