Definition:Real Number/Axioms

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Definition

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

\((\R A0)\)   $:$   Closure under addition      \(\displaystyle \forall x, y \in \R:\) \(\displaystyle x + y \in \R \)             
\((\R A1)\)   $:$   Associativity of addition      \(\displaystyle \forall x, y, z \in \R:\) \(\displaystyle \paren {x + y} + z = x + \paren {y + z} \)             
\((\R A2)\)   $:$   Commutativity of addition      \(\displaystyle \forall x, y \in \R:\) \(\displaystyle x + y = y + x \)             
\((\R A3)\)   $:$   Identity element for addition      \(\displaystyle \exists 0 \in \R: \forall x \in \R:\) \(\displaystyle x + 0 = x = 0 + x \)             
\((\R A4)\)   $:$   Inverse elements for addition      \(\displaystyle \forall x: \exists \paren {-x} \in \R:\) \(\displaystyle x + \paren {-x} = 0 = \paren {-x} + x \)             
\((\R M0)\)   $:$   Closure under multiplication      \(\displaystyle \forall x, y \in \R:\) \(\displaystyle x \times y \in \R \)             
\((\R M1)\)   $:$   Associativity of multiplication      \(\displaystyle \forall x, y, z \in \R:\) \(\displaystyle \paren {x \times y} \times z = x \times \paren {y \times z} \)             
\((\R M2)\)   $:$   Commutativity of multiplication      \(\displaystyle \forall x, y \in \R:\) \(\displaystyle x \times y = y \times x \)             
\((\R M3)\)   $:$   Identity element for multiplication      \(\displaystyle \exists 1 \in \R, 1 \ne 0: \forall x \in \R:\) \(\displaystyle x \times 1 = x = 1 \times x \)             
\((\R M4)\)   $:$   Inverse elements for multiplication      \(\displaystyle \forall x \in \R_{\ne 0}: \exists \frac 1 x \in \R_{\ne 0}:\) \(\displaystyle x \times \frac 1 x = 1 = \frac 1 x \times x \)             
\((\R D)\)   $:$   Multiplication is distributive over addition      \(\displaystyle \forall x, y, z \in \R:\) \(\displaystyle x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z} \)             
\((\R O1)\)   $:$   Usual ordering is compatible with addition      \(\displaystyle \forall x, y, z \in \R:\) \(\displaystyle x > y \implies x + z > y + z \)             
\((\R O2)\)   $:$   Usual ordering is compatible with multiplication      \(\displaystyle \forall x, y, z \in \R:\) \(\displaystyle x > y, z > 0 \implies x \times z > y \times z \)             
\((\R O3)\)   $:$   $\struct {R, +, \times, \le}$ is Dedekind complete             


These are called the real number axioms.


Sources