Course Notes for Calculus II

The historical problem of formalizing the limit is deeply connected with the problem of the continuum. There are many historical forms and presentations of the problem of the continuum, but here is a common version: how can an infinitely divisible number line exist and how can we be sure it doesn’t have any holes? The continuum is the name of this infinitely divisible and gap-less number line, so the problem is essentially a problem of giving a good, formal construction of the continuum. In modern mathematics, this is done by constructing the real numbers \(\RR\text{,}\) and proving that they are ordered in such a way as to form this infinitely divisible, gap-less continuum, which we call the real number line.

In this section, we’ll review the idea of constructing number sets, starting with \(\NN\) and working up to \(\RR\text{.}\) This is a formalizing process: intuitively, we have ideas of natural numbers, integers, rational numbers and real numbers. However, mathematics isn’t content with these intuitive ideas. It wants clear, formal definitions.

We’ll start with the assumption that the natural numbers are given. (The construction of the natural numbers out of set theory is a fascinating piece of mathematics, but it takes a lot of time so we will unfortunatley have to skip it).

The construction of \(\ZZ\) from \(\NN\) is relatively easy: it is essentially equivalent to the childhood realization that negative numbers exists and have meaning. Formally, the construction is stated in this definition.

These properties are sufficient to give the entire structure of the integers, fitting them into the existing addition and multiplication of the natural numbers. They also define subtraction, by saying that \(n-m\) is defined to be \(n + (-m)\text{.}\)

The construction of \(\QQ\) from \(\ZZ\) is also relatively easy and proceeds in a very similar way. As with \(\ZZ\text{,}\) where we defined new symbols \(-n\) and put them into our arithmetic system, here we also define new symbols and give rules to tell how the symbols fit into the existing arithmetic.

In the search for the continuum, the rational numbers are already close. They have the infinitly divisible property we want: between any two rational numbers we can construct a new rational number, simply take the sum of the two numbers divided by 2. However, they lack the gap-less nature of the continuum. This is a very old problem: the Pythagoreans had trouble with the idea that \(\sqrt{2}\text{,}\) which is a real, measurable length, wasn’t represented by a fraction. \(\sqrt{2}\) is an example of a gap in the rational numbers. The real numbers are constructed to extend \(\QQ\) by filling in the gaps.

There are a number of ways to do this. Intuitively, we’ve treated real numbers as all decimal expansions. This is a reasonable definition, but it implicitly relies on more advanced results about the convergence of infinite series. The trouble with these decimals is that they are infinite: unlike the \(-n\) symbols that defined \(\ZZ\) or the \(\frac{a}{b}\) symbols that defined \(\QQ\text{,}\) we can’t actually write down the symbols that define \(\RR\text{,}\) since they are potentially non-repeating infinite strings of digits. This is quite unsatisfying. (Unfortunately, we will eventually discover that there isn’t really any way to write the elements of \(\RR\) in any concise way.)

I’m going to present two ideas which were used historically to define \(\RR\text{.}\) The first is quite abstract, but beautiful in its approach; the second is a little bit more useful.

The first approach is due to the 19th century mathematician Dedekind and is called the method of Dedekind cuts. The idea is beautifully simple: the problem with \(\QQ\) is that it has gaps. Why not define \(\RR\) to be exactly those gaps?

First, all rational numbers provide such a cut. If I take \(\frac{3}{2}\text{,}\) I can cut \(\QQ\) into all pieces \(\lt \frac{3}{2}\) and all pieces \(\geq \frac{3}{2}\text{.}\) Notice that by starting with a rational number, I have a strict endpoint to one of my pieces: the upper piece has \(\frac{3}{2}\) as its lowest element.

Second, any gap in \(\QQ\) is a Dedekind cut. Take \(\sqrt{2}\text{,}\) for instance. I can split \(\QQ\) into those numbers which are \(\lt \sqrt{2}\) and those \(> \sqrt{2}\text{.}\) As such, \(\sqrt{2}\) defines a Dedekind cut, hence a real number in this construction. Notice there that neither piece has a lowest element in \(\QQ\text{:}\) the new ‘numbers’ are the cuts for which neither piece has a lowest or highest element.

As I said, this is a very abstract presentation of \(\RR\text{.}\) It is stange and difficult to work with. We could go on to define arithmetic on cuts, but it’s a laborious and tricky project. I’m including Dedekind cuts for their extreme elegance and style: gaps are the problems, so lets define the solution to be exactly those gaps!

The second approach is also due to a 19th century mathematician, this time Cauchy. It relates to limits and our formalization of them. (Cauchy was heavily involved and influential in the 19th century formalization of calculus and limits).

For our current purposes, we’ll leave aside the further formalization of the construction of \(\RR\) and make use of the intuitive idea of decimal expansions. What is important is that we have constructed the continuum: \(\RR\) is infinitely divisible and has no gaps.