Continuum (set theory)

In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by ${\mathfrak {c}}$ .^[1]^[2] Georg Cantor proved that the cardinality ${\mathfrak {c}}$ is larger than the smallest infinity, namely, $\aleph _{0}$ . He also proved that ${\mathfrak {c}}$ is equal to $2^{\aleph _{0}}\!$ , the cardinality of the power set of the natural numbers.

For other uses, see Continuum.

The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers, $\aleph _{0}$ , or alternatively, that ${\mathfrak {c}}=\aleph _{1}$ .^[1]

C is with respect to <.

simply ordered

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, 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)

countable

C has no first element and no last element. ()

Unboundedness axiom

According to Raymond Wilder (1965), there are four axioms that make a set C and the relation < into a linear continuum:

These axioms characterize the order type of the real number line.

Continuum (set theory)

simply ordered

Dedekind cut

countable

Unboundedness axiom

Aleph null

Suslin's problem

Transfinite number