Dedekind-Cantor Axiom of Continuity of a Real Line

Dedekind-Cantor Axiom of Continuity of a Real Line

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