Given the
D1275: Set of basic numbers $[-\infty, \infty]$, we define the following arithmetic rules:
| (1) |
\begin{equation}
\forall \, x \in (- \infty, \infty] : \infty + x = x + \infty = \infty
\end{equation}
|
| (2) |
\begin{equation}
\forall \, x \in [- \infty, \infty) : - \infty + x = x + (- \infty) = - \infty
\end{equation}
|
| (3) |
\begin{equation}
\forall \, x \in (- \infty, \infty] : \infty \cdot 0 = 0 \cdot \infty = 0
\end{equation}
|
| (4) |
\begin{equation}
\forall \, x \in [- \infty, \infty) : - \infty \cdot 0 = 0 \cdot (- \infty) = 0
\end{equation}
|
| (5) |
\begin{equation}
\forall \, x \in (-\infty, \infty) : (x < \infty) \text{ and } (- \infty < x)
\end{equation}
|
| (6) |
\begin{equation}
\infty \cdot \infty = (- \infty) \cdot (- \infty) = \infty
\end{equation}
|
| (7) |
\begin{equation}
\infty \cdot (- \infty) = - \infty \cdot \infty = - \infty
\end{equation}
|
| (8) |
\begin{equation}
- (\infty) = - \infty
\end{equation}
|
| (9) |
\begin{equation}
- (- \infty) = \infty
\end{equation}
|
In particular, expressions such as $\infty - \infty$ and $- \infty + \infty$ are left undefined (are not accepted as well-founded formulas).
