Axioms of Continuity

These axioms are the axioms which give us our correspondence between the real line and a Euclidean line. These are necessary to guarantee that our Euclidean plane is complete. The first axiom gives us some information about the relative sizes of segments as compared one to another. ARCHIMEDES' AXIOM.  If AB and CD are any segments, then there is a number n such that if segment CD is laid off n times on the ray emanating from A, then a point E is reached where and B is between A and E. This is derived from the Archimedean Axiom in the real number system. This should not be surprising, for we wish to have a one-to-one correspondence between each euclidean line and the set of real numbers. In the real line the Archimedean Postulate takes on the flavor: Archimedean Postulate: Let M and e be any two positive numbers. Then there is a positive integer n such that

The main point for geometry is that if you choose any segment to be of unit length, then every segment has finite length with respect to this measure. Nothing can be too big, ; and nothing can be too small, , where CD was chosen as our unit length.

This is still not enough for the purposes of geometry, for the set of rational numbers satisfies this property, but causes trouble in another situation. Let us consider the set of rational points in the cartesian plane. Call this . Consider the ray passing from the origin through the point (1,1). This segment has length . Now, on the ray making up the positive real axis we are unable to find a point satisfying Congruence Axiom 1. No point exists whose distance to the origin is . We require a stronger property. Dedekind's Axiom.  Suppose that the set of all points on a line is the union of two nonempty subsets such that no point of is between two points of and vice versa. Then there is a unique point O lying on such that if and only if and and . In order to better understand this axiom, we need to study the concept in the real line.

Suppose that with a<b. Suppose that satisfying

1. .
2. .
3. and .
4. If and a<y<x, then .

The fourth item above indicates that we have the property that if , then . Let . CLAIM: If z<w<b, then . Assume not, so that . This means that . This places , which then means that , a contradiction to item (2). Thus, and, in fact, .

There are three different situations possible here.

1. .
2. .
3. If neither of the above two conditions occurs, then we classify numbers as S₁-numbers or S₂-numbers. We are willing to believe that there exists an element so that if x<c then , and if x>c then . The existence of such a real number is guaranteed by the Least Upper Bound Axiom for the real numbers. Since, is bounded above by b, it has a least upper bound. Does this least upper bound separate the sets and as above? Let and let . Then, by the definition of a least upper bound, x<c and for any , . Thus, if y<c, then . If y>c then y cannot be in for c is bigger than every element in . Thus, .

The sets and form a Dedekind cut   of the set [a,b], and c is called the Dedekind number of the cut. Our Dedekind's Axiom is a translation of this phenomenon to a line. Without Dedekind's Axiom there is no guarantee that there is a segment of length or of length e, or of certain other non-constructible lengths, e.g. . It is Dedekind's Axiom that allows us to make the correspondence of the line in our geometry and the real line. There are well-defined geometries that exist without the Dedekind Axiom, such as the geometry of the surd plane. They do not have all of the properties which we wish to have, or to which we are accustomed to having. It is with Dedekind's Axiom that we are able to introduce a coordinate system and do geometry analytically, in the fashion of Fermat and Descartes.

To see why we will want Dedekind's Axiom, consider the manner in which you construct the perpendicular to a given line at a given point. First, using the given point as a center, draw a circle of positive radius. The circle intersects the line in two points. At each of these points, you then construct a circle of larger radius and these two circles intersect in two points. Drawing the line between the two points of intersection gives a line perpendicular to the given line at the given point. There are two problems with this proof, and they are both very subtle. Why does the line intersect the circle at all? Why do the two circles then intersect? In the surd plane these are not necessarily true!

The first of these problems is addressed by the following principle, which follows from Dedekind's Axiom. We define a point A to be inside a circle centered at O with radius OR if OA < OR. A point B is outside the circle if OB > OR. Elementary Continuity Principle . If one endpoint of a segment is inside a circle and the other outside, then the segment intersects the circle. We shall prove this after we have developed some theorems in geometry that we shall need.

The second problem above is that of the intersection of two circles. It is addressed by the following principle, which is again a corollary of Dedekind's Axiom. Circular Continuity Principle . If a circle has one point inside and one point outside another circle , then the two circles intersect in two points. We shall prove this later also.