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Inhaltsverzeichnis

Frontmatter

1. Alice in Logiland — in which we meet Alice, Tweedledee and Tweedledum, and Logic …

Abstract
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.
John Baylis, Rod Haggarty

2. Unique Factorisation — in which trivial arithmetic reveals a glimpse of hidden depths

Abstract
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.
John Baylis, Rod Haggarty

3. Numbers—in which we abandon logic to achieve understanding, then use logic to deepen understanding

Abstract
How and why do we abandon logic while we learn about numbers? Just think back to your first meeting with ‘one’, ‘two’, ‘three’…. These characters were probably of very minor importance initially, being no more than labels to distinguish the verses of ‘one, two, buckle my shoe’, etc. Then they began to have relationships with one another— ‘one’ for some reason always came before ‘two’; and relationships with the outside world— ‘one’, ‘two’, ‘three’ were names attached to small collections of things, but ‘ten’, ‘eleven’, ‘twelve’ were associated with bigger collections. So, even at pre-school age, we had experience of numbers being used in at least three ways—as arbitrary labels, as a means of ordering events (Monday first, Tuesday second …) and as measures (of height, weight, more numerous, less numerous…)—and these interpretations seem to have very little to do with one another. If we imagine pre-school children sufficiently precocious to ask ‘What is a number?’ it would be extremely difficult to give them a sensible answer.
John Baylis, Rod Haggarty

4. The Real Numbers—in which we find holes in the number line and pay the price for repairs

Abstract
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.
John Baylis, Rod Haggarty

5. A Variety of Versions and Uses of Induction—in which another triviality plays the lead

Abstract
tweedledee: I’ve started to educate myself, Alice, as you suggested. I found a little book in the Red Queen’s library by some chap called Fibonacci. They had very quaint ways of describing themselves in those days: this book was … ‘by Leonardo, the everlasting rabbit breeder of Pisa’.
John Baylis, Rod Haggarty

6. Permutations—in which ALICE is transformed

Abstract
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.
John Baylis, Rod Haggarty

7. Nests—in which the rationals give birth to the reals and the scene is set for arithmetic in ℝ

Abstract
Deep in conversation, Alice and the Tweedle twins have wandered into an unfamiliar part of the forest.
John Baylis, Rod Haggarty

8. Axioms for ℝ—in which we invent Arithmetic, Order our numbers and Complete our description of the reals

Abstract
The previous chapter represents the culmination of what computing aficionados might call the ‘bottom-up’ approach to the real numbers. That is to say, we began with the natural numbers ℕ and constructed, successively, the integers ℤ, the rational numbers ℚ and finally the real numbers ℝ. Now we examine the ‘top-down’ approach and adopt a more algebraic stance. The real numbers will be defined by an abstract set of axioms from which we shall deduce (as theorems), at first elementary, and later more sophisticated, properties of ℝ. The advantage of the axiomatic approach is that it does not depend on any preconceived ideas of what real numbers are. However, some of our first ‘theorems’ will appear trivial to our trained minds—the point to bear in mind is that they are consequences of even more basic assumptions— namely, our axioms. To make the following exposition more palatable, we shall present the defining axioms in three measured helpings. We shall also enlist the aid of our friends Alice, Tweedledee and Tweedledum… so, to work!
John Baylis, Rod Haggarty

9. Some Infinite Surprises—in which some wild sets are tamed, and some nearly escape

Abstract
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!
John Baylis, Rod Haggarty

10. Sequences and Series— in which we discover very odd behaviour in even the smallest infinite set

Abstract
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.
John Baylis, Rod Haggarty

11. Graphs and Continuity—in which we arrange a marriage between Intuition and Rigour

Abstract
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.
John Baylis, Rod Haggarty

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