Alice in Numberland

In Alice through the Looking-glass Alice meets Tweedledee and Tweedledum in the forest. The brothers look so much alike that Alice cannot tell them apart. In our version of the story one of the brothers tells lies on Mondays, Tuesdays and Wednesdays but tells the truth on the other days of the week. The other brother lies on Thursdays, Fridays and Saturdays but is truthful on the remaining days of the week. One day Alice meets the two in the forest, and the following conversation takes place:

FIRST BROTHER: I am Tweedledum.

SECOND BROTHER: I am Tweedledee.

ALICE: Ah, so it’s Sunday.

This chapter concerns mainly familiar old friends, the set ℕ of positive whole numbers variously known as the natural numbers or counting numbers, { 1, 2, 3, 4, …}. We think of counting as a very primitive notion firmly rooted in reality, yet already the innocent three dots in { 1, 2, 3, 4, …} may have taken us beyond reality into the realms of pure thought. The dots are usually interpreted as ‘and so on for ever’, which expresses our notion that ℕ is an infinite set. Cosmologists have still not made up their minds whether we live in a finite or an infinite universe, and in the former case there could be no such thing as an infinite set of real objects. However, mathematicians do not, in general, see this as a problem, and confidently assume that infinite sets (even if they are only sets of mental objects) can be handled with safety.

As advertised in the previous chapter, we shall take for granted the elementary arithmetic of fractions and all the generally illogical manoeuvrings it takes to get there. What we now need is a working definition of the set ℚ of rational numbers and a specific interpretation or model of them. Both are easy: the model is the familiar number line in which the rational number x is represented as a distance x along the line from 0, to the left or right, depending on whether x is negative or positive; and our (semi-formal) definition of a rational number is any number which can be expressed as the ratio between two integers, n/m, with the proviso that m is not zero.

This chapter departs from the main stream in having nothing ostensibly to do with numbers. So, if your main concern is understanding the number system, you lose nothing by skipping this chapter. If at some stage (and exactly which stage doesn’t really matter) you tackle it, we hope you will gain from it an appreciation that many of the concepts discussed in the main stream with reference to numbers are really of much wider applicability. The concepts we have in mind are factorisation, uniqueness, well-defined and equivalence classes.

Deep in conversation, Alice and the Tweedle twins have wandered into an unfamiliar part of the forest.

In this chapter we look at the properties, many of them paradoxical, of infinite sets. The subject is closely associated with the name of Georg Cantor (1845–1918). Unlike many, perhaps most, major theories in mathematics, Cantor’s ideas on infinite sets owe little to foundations built over previous centuries by other mathematicians. He was the source of most of the ideas, and for this reason the subject is relatively easy to tie down to its origins. We shall be adding a few remarks to give some historical colour to our story, but by the end of the chapter you will probably agree that the subject is quite colourful enough anyway!

Much of the effort we have expended in developing the system ℝ of real numbers and in describing the infinite can now be put to good use. In this chapter we shall be investigating limiting processes, the very foundation of analysis. We begin by agreeing that an infinite sequence is a countably infinite set of real numbers occurring in some definite order, a₁,a₂,a₃, …, a _n , …. Each a _i ∈ℝ and there is one a _i for each i∈ℕ. A favoured abbreviation for a sequence is (a _n ), where a _n denotes the nth term of the sequence.

The mathematical idea of continuity is analogous to, but not the same as, the intuitive idea of continuity which we associate with time, space or motion. We think of time as unbroken, of space as smooth with no holes and of motion as uninterrupted. The mathematician, perverse as ever, seeks to redefine this comfortable vague notion of continuity by a more useful, more precise but more troublesome definition. The real, line, which we have taken such pains to define, is deemed to be continuous. Recall that we required the completeness axiom to plug the imperceptible gaps. In this chapter we shall be mainly concerned with the notion of a ‘continuous function’. The definition of this concept is necessarily precise but it accords, most of the time, with our notion of an unbroken curve. The current theory of mathematical continuity is an abstract logical edifice which may or may not describe the way space actually is. So far, mathematicians have been able to resolve any unexpected quirks of the rigorously defined concept of a continuous function more or less to everyone’s satisfaction. One of the founders of analysis, a Catholic priest, Bernhard Bolzano (1781–1848 ), when analysing the paradoxes of the infinite, was driven to define various intuitively obvious mathematical concepts such as continuity.