Real number theory was perhaps the most significant course I took in college. There were three of us enrolled my last semester, a number small enough that that it could have been cancelled. I’m indebted to the late Prof. Jack Lamoreaux and the math department that it wasn’t. Each student would spend 10-20 minutes at the board pursuing Dedekind’s question, Was sind und was sollen die Zahlen?. Prof. Lamoreaux would guide us and involve the two sitting students with a minimal touch. For the last five minutes of class, he would lay out a direction for the next time we would meet. There were no exams, as we exposed continually all that we knew and all that we lacked knowing.
There were no assigned texts. We started with Peano’s axioms, a few simple statements that define the set of natural numbers. Using logic and set theory, we showed that the set defined by Peano’s axioms have various of the properties we think of the natural numbers having. Rules of arithmetic and comparison can be described that work like those we’ve known since childhood.
Having the set of natural numbers at our disposal, we constructed integers as classes of ordered pairs of natural numbers. We did something similar for the rational numbers, and proved, for instance, that the rational numbers are an infinite, countable set.
The next step was to define the set of real numbers based on the framework we had built up from the natural numbers. This step was qualitatively different from those that produced integers and rational numbers. My main tool for this task was the Cauchy sequence. A Cauchy sequence is an ordered set of numbers that converges. Defining where it converges is the interesting part because it may be a value of a kind that the sequence elements are not. A sequence of rational numbers can converge around something that is not a rational number, so how can one describe where they are going? Remember, the task is to define the real numbers, so their existence can’t be used until we’ve defined what they are. The way the problem is handled with a Cauchy sequence is that for any distance, no matter how small, and often labeled ε, a number N can be found such that any two elements in the sequence beyond the Nth are closer than distance epsilon apart. Classes of converging sequences of rational numbers can be used as a definition of the set of real numbers.
An example is √2, that number which when multiplied by itself equals 2. This number is not a rational number. Sequences of rational numbers can be defined, however, that come as close as desired to √2. One ancient Babylonian left behind 1 + 24/60 + 51/602 + 10/603 as a rational approximation to √2, a value that is good to 1 part in 2 million. The Babylonian method of approximating square roots converges quadratically, so the next approximation in a sequence built by that method would be good to 1 part in 11 trillion. Even if we went past the nineth number in that sequence where the approximations all match one another for the first thousand digits, or past the twentieth where they match for the first million digits, nowhere in a sequence of rational numbers could √2 be found.
None of the elements of the sequence is a real number; the sequence is the real number.
“For my thoughts are not your thoughts, neither are your ways my ways, saith the Lord. For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts than your thoughts.”-Isaiah