I think I understand the fundamental concept of infinity. Elementary mathematics define $\infty := \frac{x}{0}$, for every $x$. And also $\infty := \frac{-x}{0}$ for every $x$. I know only one definition of $-\infty$ as $-\infty= 0-(\infty)$. Is there any other way to define $-\infty$?
How to define $-\infty$?
-
20I sure hope *elementary* mathematics does **not** define infinity this way. – 2011-09-14
-
3To expand on Srivatsan's comment: Infinity is not one well-defined thing. In various areas of non-elementary mathematics, you can speak about abstractions that can be interpreted intuitively as "there's nothing finite to put here, but something different with such-and-such properties". But there are many different variants on this, and none of them claim to be _the_ infinity (as there's no such thing). Some of these concepts can be notated with "$\infty$" by convention within the field they are used in, but that's just convenient notation with limited applicability. – 2011-09-14
-
0@Srivatsan +1. I agree that infinity is not defined as such a way. I just put a commonly used formulation. – 2011-09-14
-
1Srivatsan, I sure hope infinity _can_ be defined in an elementary way. – 2011-09-14
-
0@Dan, you can probably define _an_ infinity in an elementary way, as long as you don't think your definition captures everything everyone wants to say about things that are not finite. – 2011-09-14
-
0Note that by your "definition", $-\infty = 0 - (\infty) = 0 - (-x/0) = +x/0 = \infty$. – 2011-09-14
3 Answers
Infinity is not defined in the way you described; something similar can be defined with limits but I think it is a confusing approach.
Here's a more formal definition: $\infty$ and $-\infty$ are points added to $\mathbb{R}$ in such a way the for all $a\in\mathbb{R}$ we have $-\infty < a < \infty$. Topologically speaking, open balls around $\infty$ are subsets of the form $\{x\in\mathbb{R}|x>a\}$ for a given $a$, and open balls around $-\infty$ are subsets of the form $\{x\in\mathbb{R}|x
This allows to formally define concepts like tending to $\infty$ or $-\infty$ with the usual topological approach, and the extanded $\mathbb{R}$ is still a linearly ordered set (although it is no longer a field since arithmetic involving $\infty$ will no longer preserve the nice properties it has in $\mathbb{R}$).
$-\infty$ can be defined as the surreal number $\{\emptyset|-\mathbb{N}\}$.
-
1Well that is in fact $-\omega$. Stricly speaking, $1/0$ has no solution even in the surreal numbers, nor in any other field. If we extend this to a set model with proper classes, then $-\infty = \{\emptyset|\mathbf{No}\}$ – 2012-10-09
In the real line you can "define" $\infty = \frac{1}{0^+}$ and $-\infty = \frac{1}{0^-}$, but these are really limits: $$-\infty = \lim_{x\to0^-} \frac 1x$$ Here $x\to0^-$ means that $x$ approaches $0$ from the left, i.e., using negative numbers.
-
0I missed those negative and positive zeros. :) – 2011-09-14