In the mathematical field of
set theory, the continuum means the
real numbers, or the corresponding (infinite)
cardinal number, denoted by .[1][2]Georg Cantor proved that the cardinality is larger than the smallest infinity, namely, . He also proved that is equal to , the cardinality of the
power set of the
natural numbers.
If [A,B] is a cut of C, then either A has a last element or B has a first element. (compare
Dedekind cut)
There exists a non-empty,
countable subset S of C such that, if x,y ∈ C such that x < y, then there exists z ∈ S such that x < z < y. (
separability axiom)