Definition:Real Number/Axiomatic Definition

Definition

Let $\struct {R, +, \times, \le}$ be a Dedekind complete totally ordered field.

Then $R$ is called the (field of) real numbers.

Real Number Axioms

The properties of the field of real numbers $\struct {R, +, \times, \le}$ are as follows:

 $(\R \text A 0)$ $:$ Closure under addition $\displaystyle \forall x, y \in \R:$ $\displaystyle x + y \in \R$ $(\R \text A 1)$ $:$ Associativity of addition $\displaystyle \forall x, y, z \in \R:$ $\displaystyle \paren {x + y} + z = x + \paren {y + z}$ $(\R \text A 2)$ $:$ Commutativity of addition $\displaystyle \forall x, y \in \R:$ $\displaystyle x + y = y + x$ $(\R \text A 3)$ $:$ Identity element for addition $\displaystyle \exists 0 \in \R: \forall x \in \R:$ $\displaystyle x + 0 = x = 0 + x$ $(\R \text A 4)$ $:$ Inverse elements for addition $\displaystyle \forall x: \exists \paren {-x} \in \R:$ $\displaystyle x + \paren {-x} = 0 = \paren {-x} + x$ $(\R \text M 0)$ $:$ Closure under multiplication $\displaystyle \forall x, y \in \R:$ $\displaystyle x \times y \in \R$ $(\R \text M 1)$ $:$ Associativity of multiplication $\displaystyle \forall x, y, z \in \R:$ $\displaystyle \paren {x \times y} \times z = x \times \paren {y \times z}$ $(\R \text M 2)$ $:$ Commutativity of multiplication $\displaystyle \forall x, y \in \R:$ $\displaystyle x \times y = y \times x$ $(\R \text M 3)$ $:$ Identity element for multiplication $\displaystyle \exists 1 \in \R, 1 \ne 0: \forall x \in \R:$ $\displaystyle x \times 1 = x = 1 \times x$ $(\R \text M 4)$ $:$ Inverse elements for multiplication $\displaystyle \forall x \in \R_{\ne 0}: \exists \frac 1 x \in \R_{\ne 0}:$ $\displaystyle x \times \frac 1 x = 1 = \frac 1 x \times x$ $(\R \text D)$ $:$ Multiplication is distributive over addition $\displaystyle \forall x, y, z \in \R:$ $\displaystyle x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$ $(\R \text O 1)$ $:$ Usual ordering is compatible with addition $\displaystyle \forall x, y, z \in \R:$ $\displaystyle x > y \implies x + z > y + z$ $(\R \text O 2)$ $:$ Usual ordering is compatible with multiplication $\displaystyle \forall x, y, z \in \R:$ $\displaystyle x > y, z > 0 \implies x \times z > y \times z$ $(\R \text O 3)$ $:$ $\struct {R, +, \times, \le}$ is Dedekind complete

These are called the real number axioms.

Also defined as

Some sources additionally specify that $\struct {R, \le}$ be densely ordered.

This condition, while conceptually important, is superfluous, by Totally Ordered Field is Densely Ordered.