Arithmetic structure of real numbers - Axioms of Choice

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Arithmetic structure of real numbers

Set

definiendum

$\langle \mathbb R,+_\mathbb{R},\cdot_\mathbb{R} \rangle$

postulate	$r +_\mathbb{R} s = \{q+_\mathbb{Q}p\ \|\ (q\in r)\land (p\in s)\}$
postulate	$r -_\mathbb{R} s = \{q-_\mathbb{Q}p\ \|\ (q\in r)\land (p\in \mathbb Q\setminus s)\}$
postulate	$-_{\mathbb R}r = \{q-_\mathbb{Q}p\ \|\ (p\in \mathbb Q\setminus r)\land (q<0)\}$

postulate	$r\ge 0\land s\ge 0\implies r\cdot_\mathbb{R}s = \{q\cdot_\mathbb{Q}p\ \|\ (q\in r)\land (p\in s)\land (q,p\ge 0)\}\cup\{q\ \|\ (q\in\mathbb Q)\land (q<0)\}$
postulate	$r\ge 0\land s < 0\implies r\cdot_\mathbb{R}s = -(r\cdot_\mathbb{R}(-s))$
postulate	$r < 0\land s\ge 0\implies r\cdot_\mathbb{R}s = -((-r)\cdot_\mathbb{R}s)$
postulate	$r < 0\land s < 0\implies r\cdot_\mathbb{R}s = (-r)\cdot_\mathbb{R}(-s)$

postulate	$r\ge 0\land s > 0\implies r/_\mathbb{R}s = \{q/_\mathbb{Q}p\ \|\ (q\in r)\land (p\in \mathbb Q\setminus s)\}$
postulate	$r\ge 0\land s < 0\implies r/_\mathbb{R}s = -(r/_\mathbb{R}(-s))$
postulate	$r < 0\land s > 0\implies r/_\mathbb{R}s = -((-r)/_\mathbb{R}s)$
postulate	$r < 0\land s < 0\implies r/_\mathbb{R}s = (-r)/_\mathbb{R}(-s)$

The operations $+_\mathbb{Q}$ and $\cdot_\mathbb{Q}$ on the right hand sides are these of arithmetic structure of rational numbers.

Discussion

Reference

Wikipedia: Real number, Construction of the real numbers

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