...continuity is contained in the following axiom : "If all the points of a straight line are divided into two classes such that every point of the first...class lies to the left of every point of the second, then there exists one point and only one which effects this division of all points into two classes,...

...different form (SMigkeit und irrationale Zahlen, 1872, new edition 1905, p. u). " If all points of a straight line fall into two classes such that every...lies to the left of every point of the second class, there exists one and only one point which produces this division of all the points into two classes,...

...does exist: and that the first class of points does include points besides x = 0. Then it is clear that every point of the first class lies to the left of every point of the second class. Hence there is a point A (x = R, say) which divides the two classes, and may itself belong to either...

...statement Dedekind's view of the irrational as a "schnitt," or cut, may be given in his own words, "If all points of the straight line fall into two classes, such that every point '7 it 22 y /A 2/ 20 /J 6 20 /siS Fig. 33 — THE NINE SECTIONS OF A MAGIC CURE. (Andrews.) Fig. 34...

...the fact that when all the points of a straight line are divided into two classes, in such a manner that every point of the first class lies to the left of every point of the second class, there exists one and only one point of the straight line which produces this division of all the points,...

...all the points of a straight line fall into two classes so that every point of the first class falls to the left of every point of the second class, then there exists one and only one point thus severing the straight line into two portions and dividing all the points into two classes. (Dedekind,...

...Braunschweig 1872, 10). [If the points of a line are divided into two classes, in such a way that each point of the first class lies to the left of every...second class, then there exists one and only one point of division which produces this particular subdivision into two classes, this cutting of the line into...

...continua, about such a construction: "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 lties to the left [our emphasis] of every point of the second class, then there exists one and only...

...(1831-1916) introduced into geometry the following continuity postulate: // all points of a horizontal straight line fall into two classes, such that every...the left of every point of the second class, then FIGURE 2.5 there exists one and only one point that produces this division of all points into two classes...

...einen Punkt geteilte Gerade zitiert Heath ebd. S.236 in der Formulierung Dedekinds: "If all poinis of a straight line fall into two classes such that every...lies to the left of every point of the second class, there exists one and only one point which produces this division of all the points into two classes,...