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 \mathbb{R}, we find that there are no gaps in \mathbb{R} of the kind \mathbb{Q} has. We may, therefore, say that the real numbers form a continuous system. On account of this characteristic, \mathbb{R} 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.