We have been continuing our focus on the foundations of mathematics. We have discussed the basic ideas of set theory and logic, proof techniques, quantifiers, and have begun to develop the number system. We have laid the groundwork for discussing the real numbers by exploring sequences, functions, and relations. Last column we discussed how to form infinite sets.
In this column we will explore some geometric ideas that will help us as we move ahead, and I will introduce the concept of real numbers.
We need to understand that there are some ideas that will be vague to begin with. These are necessary because we want to avoid circular reasoning. These are not formal definitions, we simply agree to understand them within the context they are presented. Such ideas are called primitive.
Any object that has no size or shape is called a point.
Any object that has infinite length without width of height is a line.
We can now make some postulates about points and lines. The shortest line passing through two points is called a straight line. Through any two points there can be only one straight line. There are at least two points on every line. We can continue like this and create geometry.
We can also define two points on the same line as colinear.
Imagine a straight line before us. This implies that we can define two points on it such that the distance between these two points is a minimum. Of course we have yet to define distance or minimum, but we are familiar with these words. Given any two colinear points, A and B, there is a unique quantity AB called the distance between them. We understand the smallest possible distance between our points is the minimum distance.
Let's say that we have the points A and B, now slip a point C somewhere in the distance AB. We now have two distances, AC and CB,
We say that C is between A and B. If we keep doing this we notice something, we can always slip another point between any two existing points. In this way we can see that a line is an infinitely large collection of points.
Another way of thinking about it, based on a previous column ("Sequences of Numbers"), is that the process of getting smaller distances by adding new points between existing points forms a Cauchy sequence. What we have done is to partition the line into a set of equivalence classes, each defined by a Cauchy sequence (see the column "Functions and Relations").
If we have the set of all Cauchy sequences of rational numbers, 
, such that
we can define the set of equivalence classes of the elements of 
with the symbol 
. We say that 
is the set of real numbers (or just the reals) and 
is a real number.
The set of reals can be represented by a line.
1. Prove that given three points on a line one must be between the other two.
2. Explain the line of reasoning behind the statement, "...the process of getting smaller distances by adding new points between existing points forms a Cauchy sequence."
3. Justify the Ruler Axiom.
Converted by Mathematica (February 6, 2003)