Set of basic numbers

Let $\mathbb{R}$ be the D282: Set of real numbers.
The set of basic numbers is the D11: Set $$[-\infty, \infty] : = \mathbb{R} \cup \{ - \infty, + \infty \}$$
Also known as
Set of extended real numbers
Conventions
Convention 0 (Real arithmetic rules involving infinities) : Given the D1275: Set of basic numbers $[-\infty, \infty]$, we define the following arithmetic rules:
 (1) $$\forall \, x \in (- \infty, \infty] : \infty + x = x + \infty = \infty$$ (2) $$\forall \, x \in [- \infty, \infty) : - \infty + x = x + (- \infty) = - \infty$$ (3) $$\forall \, x \in (- \infty, \infty] : \infty \cdot 0 = 0 \cdot \infty = 0$$ (4) $$\forall \, x \in [- \infty, \infty) : - \infty \cdot 0 = 0 \cdot (- \infty) = 0$$ (5) $$\forall \, x \in (-\infty, \infty) : (x < \infty) \text{ and } (- \infty < x)$$ (6) $$\infty \cdot \infty = (- \infty) \cdot (- \infty) = \infty$$ (7) $$\infty \cdot (- \infty) = - \infty \cdot \infty = - \infty$$ (8) $$- (\infty) = - \infty$$ (9) $$- (- \infty) = \infty$$
In particular, expressions such as $\infty - \infty$ and $- \infty + \infty$ are left undefined (are not accepted as well-founded formulas).
Child definitions