Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Binary endorelation
Preordering relation
Partial ordering relation
Ordering relation
Ordered set
Dedekind cut
Set of real numbers
Set of basic numbers
Formulation 1
Let $\mathbb{R}$ be the D282: Set of real numbers.
The set of basic numbers is the D11: Set \begin{equation} [-\infty, \infty] : = \mathbb{R} \cup \{ - \infty, + \infty \} \end{equation}
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) \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).
Child definitions
» D1699: Basic number
» D4517: Set of euclidean basic numbers