These works of G. Cantor and Dedekind were of the greatest importance in the progress of the subject.
Dedekind and Peano have worked out such ordinal theories of the number concept.
An open series is continuous if it is compact and possesses the Dedekind property.
Thus the definitions of compactness and of the Dedekind property can be at once transferred to a closed series.
The last axiom of order is that there exists at least one straight line for which the point order possesses the Dedekind property.
It follows from axioms 1-12 by projection that the Dedekind property is true for all lines.
The Dedekind property holds for the order of the points on any straight line.