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 , 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.