Definition:Real Number/Axiomatic Definition
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Definition
Let $\struct {R, +, \times, \le}$ be a Dedekind complete 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 | \(\ds \forall x, y \in \R:\) | \(\ds x + y \in \R \) | |||||
\((\R \text A 1)\) | $:$ | Associativity of addition | \(\ds \forall x, y, z \in \R:\) | \(\ds \paren {x + y} + z = x + \paren {y + z} \) | |||||
\((\R \text A 2)\) | $:$ | Commutativity of addition | \(\ds \forall x, y \in \R:\) | \(\ds x + y = y + x \) | |||||
\((\R \text A 3)\) | $:$ | Identity element for addition | \(\ds \exists 0 \in \R: \forall x \in \R:\) | \(\ds x + 0 = x = 0 + x \) | |||||
\((\R \text A 4)\) | $:$ | Inverse elements for addition | \(\ds \forall x: \exists \paren {-x} \in \R:\) | \(\ds x + \paren {-x} = 0 = \paren {-x} + x \) | |||||
\((\R \text M 0)\) | $:$ | Closure under multiplication | \(\ds \forall x, y \in \R:\) | \(\ds x \times y \in \R \) | |||||
\((\R \text M 1)\) | $:$ | Associativity of multiplication | \(\ds \forall x, y, z \in \R:\) | \(\ds \paren {x \times y} \times z = x \times \paren {y \times z} \) | |||||
\((\R \text M 2)\) | $:$ | Commutativity of multiplication | \(\ds \forall x, y \in \R:\) | \(\ds x \times y = y \times x \) | |||||
\((\R \text M 3)\) | $:$ | Identity element for multiplication | \(\ds \exists 1 \in \R, 1 \ne 0: \forall x \in \R:\) | \(\ds x \times 1 = x = 1 \times x \) | |||||
\((\R \text M 4)\) | $:$ | Inverse elements for multiplication | \(\ds \forall x \in \R_{\ne 0}: \exists \frac 1 x \in \R_{\ne 0}:\) | \(\ds x \times \frac 1 x = 1 = \frac 1 x \times x \) | |||||
\((\R \text D)\) | $:$ | Multiplication is distributive over addition | \(\ds \forall x, y, z \in \R:\) | \(\ds x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z} \) | |||||
\((\R \text O 1)\) | $:$ | Usual ordering is compatible with addition | \(\ds \forall x, y, z \in \R:\) | \(\ds x > y \implies x + z > y + z \) | |||||
\((\R \text O 2)\) | $:$ | Usual ordering is compatible with multiplication | \(\ds \forall x, y, z \in \R:\) | \(\ds x > y, z > 0 \implies x \times z > y \times z \) | |||||
\((\R \text O 3)\) | $:$ | $\struct {\R, +, \times, \le}$ 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.
Sources
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers