Axiom:Real Number/Axioms

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Definition

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.

Sources

2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers

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1957: Tom M. Apostol: Mathematical Analysis ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-3}$ Order properties of real numbers
1967: Michael Spivak: Calculus ... (previous) ... (next): Part $\text I$: Prologue: Chapter $1$: Basic Properties of Numbers: $(\text P 6)$
1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.3$: Arithmetic

\((\R \text A 0)\)	$:$	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

Axiom:Real Number/Axioms

Definition

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