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We shall assume as axioms (i.e., without proof) the following simple properties of real numbers. (The reader is certainly familiar with these properties from school algebra, where they are often regarded as “obvious”, so that it might seem superfluous to mention them. We must, however, state them as axioms

in accordance with our introductory remarks made in §1. Each axiom has a name given in parenthesis.)

A. Axioms of addition and multiplication.

I (Closure law)The sum x+y and the product xy of any two real numbers

x and y are themselves real numbers. In symbols: (_∀x, y_∈E1) (x+y)_∈E1, (xy)_∈E1.

II (Commutative laws) (_∀x, y _∈E1) x+y =y+x, xy =yx.

III (Associative laws) (_∀x, y, z _∈E1_{) (x+y) +z =x+ (y+z), (xy)z =x(yz).}

IV (Existence of neutral elements)

(a) There exists a (unique) real number, called “zero” (0), such that,

for all real x, x+ 0 =x.

(b) There exists a (unique) real number, called “one” (1), such that

1₆= 0 and, for all real x, x_·1 =x. In symbols: (∃! 0∈E1) (∀x∈E1) x+ 0 =x,

(∃! 1∈E1) (∀x∈E1) x·1 =x, 16= 0.

The numbers 0 and 1 are called the neutral elements of addition and multiplication, respectively.

V (Existence of inverses)

(a) For every real number x, there is a (unique) real number, denoted

−x, such that x+ (₋x) = 0.

(b) For every real numberx, other than0, there is a (unique)real num- ber denoted x−1_{, such that x·x}−1 _{= 1. In symbols:}

(∀x∈E1) (∃!−x∈E1) x+ (−x) = 0, (_∀x_∈E1 _|x₆= 0) (_∃!x−1 _∈E1_{) x·x}−1 _{= 1.}

§2. Axioms of an Ordered Field 53

The numbers₋xandx−1 _{are called, respectively, theadditive inverse (or}

the symmetric) and the multiplicative inverse (or thereciprocal) of x.

VI (Distributive law) (_∀x, y, z _∈E1_{) (x+y)z =xz+yz.}

Note. The uniqueness assertions in Axioms IV and V could actually be omitted since they can be proved from other axioms.

B. Axioms of order.

VII (Trichotomy) For any real numbers x and y, we have either x < y or y < x or x=y, but never two of these relations together.

VIII (Transitivity) If x, y, z are real numbers, with x < y and y < z, then x < z. In symbols:

(_∀x, y, z_∈E1) x < y < z implies x < z.

IX (Monotonicity of addition and multiplication) (a) (_∀x, y, z _∈E1_{) x < y implies x+z < y+z.}

(b) (_∀x, y, z _∈E1_{) x < y and 0< z implies xz < yz.}

Note 1. As has already been mentioned, one additional (10th) axiom will be stated later.

Note 2. While every real number has an additive inverse (Axiom V(a)), only nonzero numbers have reciprocals. The number 0 has no reciprocal. (Ax- iom V(b).)

Note 3. Note the restriction 0 < z in Axiom IX(b). It is easy to see that without this restriction the axiom would be false. For example, we have 2<3, but 2(₋1) is not less than 3(₋1). No such restriction occurs in Axiom IX(a).

Due to the introduction of inequalities “<” and the Axioms VII–IX, the real numbers may be regarded as given in some definite order, under which smaller numbers precede the larger ones. (This is why Axioms VII–IX are called “axioms of order”.) We express this fact briefly by saying thatE1 is an ordered set. More precisely, an ordered set is a set in which a certain relation “<” has been defined in such a manner that the trichotomy and transitivity laws are satisfied.

The ordering of real numbers can be visualized by “plotting” them as points on a directed line (“the real axis”), as shown below in Figure 8:

−2 ₋11_{2 −}1 ₋1₂ 0 1₂ 1 2

Therefore, real numbers are also often referred to as “points” of the real axis. We say, e.g., “the pointx” instead of “the numberx.” We assume that the reader is familiar with this process of geometric representation of real numbers. We shall not dwell on its justification since it will only be used as illustration, not as proof.

It should be noted that the axioms only specify certain properties of real numbers without indicating what these numbers actually are. This question is left entirely open, so that we may regard real numbers as just any mathematical objects that are only supposed to satisfy our axioms but otherwise areentirely arbitrary. This makes our theory more general. Indeed, our theory also applies to any other set of objects (numbers or not numbers), provided only that they satisfy our axioms with respect to a certain relation of order (<) and certain operations (+) and (_·), which may, but need not, coincide with ordinary number addition and multiplication. Whatever follows logically from the axioms must be true not only for real numbers but also for any other set that conforms with these axioms. In this connection, we introduce the following definitions.

Definition 1.

A field F is any set of objects with two operations (+) and (_·) (usually called “addition” and “multiplication”) defined in it, provided that these objects and operations satisfy the first six axioms (I–VI) listed above.

If this set is also equipped with an order relation (<) satisfying the additional three axioms VII–IX, it is called an ordered field.

In particular,the real number systemE1 _{is an ordered field. Of course,}

when speaking of ordered fields in general, the term “real number” in the axioms should be replaced by “element of F.” Similarly, 0 and 1 should be interpreted as elements of the field satisfying Axiom IV(a) and (b), but not necessarily as ordinary numbers.

E1 _{is not the only ordered field known in mathematics. Indeed, many ex-}

amples of ordered and unordered fields are studied in higher algebra. We shall encounter some of them later.

As has been mentioned, everything that can be deduced from Axioms I– IX applies not only to E1 _{but also to any other ordered field F (since F is}

supposed to satisfy these axioms). Therefore, we shall henceforth formulate our definitions and theorems in a more general way, speaking of “ordered fields” in general instead of E1 _{alone. Of course, whatever we say about ordered fields}

applies in particular to E1_{, and this particular example should be always kept}

in mind.

Definition 2.

An element x of an ordered field F is said to be positive or negative according as x > 0 or x <0. The element 0 itself is neither positive nor negative.

§2. Axioms of an Ordered Field 55

Here and henceforth “x > y” means the same as “y < x”. We also write “x≤y” for “x < y or x=y”; similarly for x≥y.

The numbers 0 and 1 have been introduced in Axiom IV, but we do not yet “officially” know what such symbols as 2, 3, 4, . . ., etc. mean, since they have not yet been defined. Indeed, we have only introduced the notion of real number, but not that of natural number (orinteger). Therefore, in our system, the latter must be defined in terms of our primitive concepts. Since, however, addition is already known, we can use it to define positive integers step by step, as follows:

2 = 1 + 1, 3 = 2 + 1, 4 = 3 + 1, 5 = 4 + 1, etc.

If this process is continued indefinitely, we obtain what is called the set of all “positive integers” (or “natural numbers”). We may say that a natural number is one that can be obtained from 0 by adding to it 1 a finite number of times. A similar process is, of course, possible not only in E1 _{but in any field. Thus}

we may speak of “natural elements” in any field.

This may serve as a preliminary definition of natural numbers. A more exact definition will be given in §5.

Definition 3.

Given several elements a, b, c, d of a field F, we define

a+b+c= (a+b) +c, a+b+c+d= (a+b+c) +d, etc. Similarly for multiplication.