Continuum (set theory)

In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by formula_1. Georg Cantor proved that the cardinality formula_1 is larger than the smallest infinity, namely, formula_3. He also proved that formula_1 is equal to formula_5, the cardinality of the power set of the natural numbers.
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, formula_3, or alternatively, that formula_7.
Linear continuum.
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.