Just Curious

Continuity of a line

I have just finished reading Professor Richard Dedekind's monumental essay on the theory of numbers, "Contunuity and Irrational Numbers", published in 1872.

In the essay the author defined continuity of a line, call L, as follows:
"I find the essence of continuity in the...following principle:
If all points of the straight line fall into two classes such that every point of the first class (call A1) lies to the left of every point of the second class (call A2), then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions."

The author claimed that in making this statement, he was "utterly unable to adduce any proof of its correctness, nor has any one the power. The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we find continuity in the line."

It would be much appreciated if any one could let me know whether this statement is the formal definition of continuity (in fact, Dedekind derived the familiar epsilon-delta definition of continuity from the aforementioned concept of "cutting", famously known as the Dedekind cut); whether it could/should be treated as an axiom in geometry; and if not, whether it is provable as a proposition.

posted by James Chang @ 2:10 PM