Axioms of the real line

Don't take these axioms too seriously! Math is not about axioms, despite what some people say. Axioms are one way to think precisely, but they are not the only way, and they are certainly not always the best way. Also, there are a number of ways to phrase these axioms, and different books will do this differently, but they are all equivalent (unless the book author was really sloppy).

The real line R has two special elements, 0, and 1, and several operations. There are two binary operations, + and *; two unary
operations, negation - and inversion x^-1, (with inversion only defined for x != 0); and an order relation <.

x*y is usually abbreviated xy.

The real line satisfies the following axioms.

I. Field axioms. For all real numbers x,y,z, we have

Additive axioms:

x+y is real (closure)
x+y=y+x (commutativity)
(x+y)+z = x+(y+z) (associativity)
x+0 = 0+x = x (identity)
x + (-x) = (-x) + x = 0 (inverse)

Multiplicative axioms:

xy is real (closure)
xy=yx (commutativity)
(xy)z = x(yz) (associativity)
x1 = 1x = x (identity)
if x !=0, then x(x^-1) = (x^-1) x = 1 (inverse)

Distributive laws: x(y+z) = xy + xz and (y+z)x = yx + zx.
Remark: Some of these laws are actually redundant, but traditionally we include them all for the symmetric effect.

II. Order axioms.

Trichotomy: For any real numbers x, y, exactly one of the following three statements is true: x=y, x>y, or y>x.
Transitivity: For any real numbers x, y, z, if one has x > y and y > z, then one necessarily has x > z.
Additive compatibility: If x,y,z are real numbers, and x > y, then x + z > y + z.
Multiplicative compatibility: If x,y,z are real numbers, and x > y and z > 0, then xz > yz.
Remark: We write x <= y to denote the statement that either x < y or x = y.

III. Least upper bound axiom.

Definition: A number M is said to be an upper bound for a set X of real numbers if one has x <= M for all x in X.
Definition: A number M is said to be a least upper bound for a set X of real numbers if it is an upper bound for X, and it is less than all other upper bounds for X.
Least upper bound axiom: If a set X has at least one upper bound, then it has a least upper bound.
Remark: This least upper bound is denoted sup(X) or lub(X).
This axiom is also called Dedekind's completeness axiom. It is the key axiom which distinguishes the real line {\bf R} from other ordered fields such as the rationals {\bf Q}.