# Show that the real field $\mathbb{R}$ has a unique ordering and indicates that ordering

Problem: Show that the real field $$\mathbb{R}$$ has a unique ordering and indicates that ordering.

My question: We knew that $$\le$$ is an ordering on $$\mathbb{R}$$. Do we need to prove that $$\le$$ is an ordering on $$\mathbb{R}$$? How to show that the uniqueness? Thank all!

EDIT 1: Suppose $$\le$$ be an ordering on $$\mathbb{R}$$, so it satisfies $$\le$$ is a total order relation on $$\mathbb{R}$$, $$\forall z \in \mathbb{R}, x \le y \Rightarrow x + z \le y + z$$, $$0 \le x, 0 \le y \Rightarrow 0 \le xy$$. Suppose $$<$$ be another ordering on $$\mathbb{R}$$ then $$x defined by $$x \le y$$ and $$x \ne y$$. Hence there is a unique ordering on $$\mathbb{R}$$.

• This looks like fairly hard work. Presumably you require that the ordering satisfies certain conditions such as (1) $0<1$ and (2) $x<y$ implies $x+z<y+z$, and (3) $x<y$ and $0<a\le b$ implies $ax<by$. Then you show this must be the same as the usual ordering on $\mathbb{Z}$, then on $\mathbb{Q}$, and finally on $\mathbb{R}$. Jan 11, 2020 at 14:17
• For the requirement indicates the ordering, it is $\le$ and I do not realize how to indicate?
– Minh
Jan 11, 2020 at 14:19
• $\le$ and $<$ are closely related: if $\le$ is defined, then $a<b$ is just shorthand for $a\le b$ and $a\ne b$. Jan 11, 2020 at 14:21
• The real numbers has this amazing property: If $a \in \mathbb R$, the either there is $b$ with $a=b^2$ or there is $b$ with $a=-b^2$. Jan 11, 2020 at 14:36

Denote by $$\leq$$ the usual ordering on $$\mathbb{R}$$ and suppose $$\preceq$$ is some ordering on $$\mathbb{R}$$. So $$(\mathbb{R},\preceq)$$ is a linearly ordered field. Suppose that $$x \neq 0$$ is a real number. Then we can prove that $$0 \preceq x^2$$ and since $$x\neq 0$$, you deduce that $$0\prec x^2$$ and also $$-x^2\prec 0$$. As a consequence we derive that $$[0,+\infty) = \{x\in \mathbb{R}\,|\,0\preceq x\}$$ Thus orderings $$\leq$$ and $$\preceq$$ have the same subset of nonnegative elements. This implies that they are the same.