Formal definition of $\mathbb{R}^+$ and total order

Question

I am looking for a formal definition of the subset of positive real number $\mathbb{R}^+$.

I know the definition of real number and I'm trying to define over $\mathbb{R}$ the symbol $\le$ without the lexicographical order.

This is my idea.

Definition of $\mathbb{R}^+$

$\mathbb{R}^+$ is a subset of $\mathbb{R}$ such that

$\\ \bullet\ 0\notin\mathbb{R}^{+};\\ \\ \bullet\ 1\in\mathbb{R}^{+};\\ \\ \bullet\ x, y\in\mathbb{R}^{+}\iff x+y\in\mathbb{R}^{+}\wedge x\cdot y\in\mathbb{R}^+.$

If the definition is correct, than I can define $\le$ as follow:

$$a\le b\iff b-a\in\mathbb{R}^{+}\cup\left\{0\right\}$$

and using the axioms that define the real set $\mathbb{R}$ I can show that it is a totally ordered set under $\le$.

My question: is the definition of $\mathbb{R}^+$ correct? Thanks.

Edit: as suggested, the axioms are insufficient to define uniquely $\mathbb{R}^+$. I have to add an axiom:

$\bullet$ If $x\in\mathbb{R}\setminus\{0\}\implies x\cdot x\in\mathbb{R}^+$

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Definition of $\mathbb{R}^+$.

$\mathbb{R}^+$ is a subset of $\mathbb{R}$ such that:

1. $0, -1\notin\mathbb{R}^+$;
2. If $x\in\mathbb{R}-\{0\}\implies x^2\in\mathbb{R}^+$;
3. $x, y\in\mathbb{R}^+\iff x+y\in\mathbb{R}^+\wedge x y\in\mathbb{R}^+$

Answer 1

A key property of the real numbers is:

A real number is nonnegative if and only if it is a square

As an aside, it turns out that in any orderable ring, the numbers that are nonnegative in every ordering of the ring are precisely those numbers that can be written as a sum of squares.