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AXIOMATIC FOUNDATIONS OF THE REAL NUMBER SYSTEM

The number system can be built up logically, starting from a basic set of axioms or ‘‘self-evident’’

truths, usually taken from experience, such as statements 1–9, Page 2.

If we assume as given the natural numbers and the operations of addition and multiplication

(although it is possible to start even further back with the concept of sets), we ?nd that statements 1

through 6, Page 2, with R as the set of natural numbers, hold, while 7 through 9 do not hold.

Taking 7 and 8 as additional requirements, we introduce the numbers                       1;    2;    3; . . . and 0. Then

by taking 9 we introduce the rational numbers.

Operations with these newly obtained numbers can be de?ned by adopting axioms 1 through 6,

where R is now the set of integers. These lead to proofs of statements such as ً2قً3ق ¼ 6, ً4ق ¼ 4,

ً0قً5ق ¼ 0, and so on, which are usually taken for granted in elementary mathematics.

We can also introduce the concept of order or inequality for integers, and from these inequalities for

rational numbers.           For example, if a, b, c, d         are positive integers, we de?ne a=b > c=d             if and only if

ad > bc, with similar extensions to negative integers.

Once we have the set of rational numbers and the rules of inequality concerning them, we can order

them geometrically as points on the real axis, as already indicated.                      We can then show that there are

points on the line which do not represent rational numbers (such as                           2,    , etc.).       These irrational

numbers can be de?ned in various ways, one of which uses the idea of Dedekind cuts (see Problem 1.34).

From this we can show that the usual rules of algebra apply to irrational numbers and that no further

real numbers are possible.