1.1 Axioms of an Ordered field

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: \begin {align*} (\forall x,y \in \mathbb {R}^1) \quad (x+y ) \in \mathbb {R}^1, (xy) \in \mathbb {R}^1 \end {align*}

II

(Commutative Law) \((\forall x,y \in \mathbb {R}^1)\), \(x+y = y+x\), \(xy = yx\).

III

(Associative Laws) \((\forall x,y,z \in \mathbb {R}^1)\), \((x+y) +z = x + (y+z)\), \((xy)z =x(yz)\)

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 \(1 \neq 0\) and, for all real \(x\), \(x \cdot 1 =x\). In symbols: \begin {gather*} (\exists ! \, 0 \in \mathbb {R}^1) \: (\forall x \in \mathbb {R}^1) \;\; x+0=x,\\ (\exists ! \, 1 \in \mathbb {R}^1) \: (\forall x \in \mathbb {R}^1) \;\; x \cdot 1 =x, \quad 1 \neq 0 \end {gather*}

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 number \(x\), other than 0, there is a (unique) real number denoted \(x^{-1}\), such that \(x \cdot x^{-1} =1\). In symbols: \begin {gather*} (\forall x \in \mathbb {R}^1) \;(\exists ! \, -x \in \mathbb {R}^1) \;\; x + (-x) =0,\\ (\forall x \in \mathbb {R}^1 \,|\, x \neq 0) \; (\exists ! \, x^{-1} \in \mathbb {R}^1) \;\; x\cdot x^{-1} =1. \end {gather*}

The numbers \(-x\) and \(x^{-1}\) are called, respectively, the additive inverse (or the symmetric) and the multiplicative inverse (or the reciprocal) of \(x\).

VI

(Distributive Law) \( (\forall x,y,z \in \mathbb {R}^1) \; (x+y)z = xz +yz.\)

Note: The Uniqueness assertions in Axioms IV and V can be proven from other axioms.

B. Axioms of order

VII

(Trichotomy) For any real numbers \(x\) and \(y\), we have either \(x<y\), \(x> y\) 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: \begin {align*} (\forall x,y,z \in \mathbb {R}^1) \;\:\: x<y<z \implies \; x<z \end {align*}

IX

(Monotonicity of addition and multiplication)

a)

\((\forall x,y,z \in \mathbb {R}^1) \; x< y\) implies \(x+z<y+z\).

b)

\((\forall x,y,z \in \mathbb {R}^1) \; x<y\) and \(z>0\) implies \(xz <yz\)

Due to the introduction of inequalities ”\(<\)”, and axioms VII-IX, the real numbers can be regarded as given in some definite order, under which smaller numbers precede the larger ones. Any set, in which a certain relation ”\(>\)” has been defined so that Trichotomy and Transitivity Laws are satisfied, is called an ordered set. \(\mathbb {R}^1\) is an ordered set.

It should be noted that the axioms only specify certain properties of real numbers without indicating what these numbers are. This question is left entirely open, so that we may regard real numbers as just any other mathematical objects that are only supposed to satisfy our axioms but otherwise are entirely arbitrary. Whatever follows from the axioms must be true not only for real numbers but also for any other set that conforms with these axioms.