# Dedekind Cantor Axiom of Continuity of a Real Line

__Dedekind Cantor Axiom of Continuity Real Line__:

To every real number corresponds a unique point of a directed straight line and conversely to every point on this straight line corresponds a unique real number.

Precisely we express the above axiom by saying that there is a one–one (1—1) correspondence between the real numbers and the points of a directed straight line. With this analogy we shall use the word point for real number and the directed straight line

may be referred to as the real line.

__Arithmetical and Geometrical Continuums__:

In view of the completeness axiom in , we find that there are no gaps in of the kind has. We may, therefore, say that the real numbers form a continuous system. On account of this characteristic, is also called the Arithmetical Continuum.

As provided by the Dedekind-Cantor axiom, we find that the systems of points on a directed straight line also do not posses gaps and, therefore, the system of points on a directed straight line is called the Geometrical Continuum.