# 実数の公理

## Statement

### 実数の公理

Defined $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$
Defined $g:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$
"$f(a,b)$" is expressed by "$a+b$"
"$g(a,b)$" is expressed by "$ab$"
Defined $\le$.
("$a\le b$" can be expressed by "$b\ge a$")
("$a\le b,a\neq b$" can be expressed by "$a\lt b$")
1. $a+b=b+a$
2. $(a+b)+c=a+(b+c)$
3. $\exists 0\in\mathbb{R},\forall a\in\mathbb{R },a+0=0+a=a$
4. $\forall a\in\mathbb{R},\exists b\in\mathbb{R},a+b=b+a=0$ ("$b$" is expressed by "$-a$")
5. $ab=ba$
6. $(ab)c=a(bc)$
7. $\exists 1\in\mathbb{R},\forall a\in\mathbb{R },a1=1a=a$
8. $\forall a\in\mathbb{R}^*,\exists b\in\mathbb{R},ab=ba=1$ ("$b$" is expressed by "$\frac{1}{a}$" or "$a^{-1}$") ( $\mathbb{R}^*=\mathbb{R} \backslash \{0\}$ )
9. $(a+b)c=ac+bc,a(b+c)=ab+ac$
10. $0\neq 1$
11. $\forall a\in\mathbb{R}\Rightarrow a\le a$
12. $a\le b, b\le a\Rightarrow a=b$
13. $a\le b, b\le c\Rightarrow a\le c$
14. $\forall a,b\in\mathbb{R}, a\le b$ or $b\le a$
15. $a\le b\Rightarrow a+c \le b+c$
16. $a\le b, c>0\Rightarrow ac\le bc$
17. $A \subset \mathbb{R}, A \neq \emptyset,$
$\exists b \in \mathbb{R}, \forall x \in A, x < b$
then
$\exists s \in \mathbb{R}, s = \mathrm{sup} A$ (連続の公理)