What is Infinity?

It depends on the context…

##bb +-## Infinity and limits
Consider the set of Real numbers ##RR##, often pictured as a line with negative numbers on the left and positive numbers on the right. We can add two points called ##+oo## and ##-oo## that do not quite work as numbers, but have the following property:

##AA x in RR, -oo < x < +oo## Then we can write ##lim_(x->+oo)## to mean the limit as ##x## gets more and more positive without upper bound and ##lim_(x->-oo)## to mean the limit as ##x## gets more and more negative without lower bound.
We can also write expressions like:

##lim_(x->0+) 1/x = +oo##
##lim_(x->0-) 1/x = -oo##

…meaning that the value of ##1/x## increases or decreases without bound as ##x## approaches ##0## from the ‘right’ or ‘left’.
So in these contexts ##+-oo## are really shorthand to express conditions or results of limiting processes.
Infinity as a completion of ##RR## or ##CC##
The projective line ##RR_oo## and Riemann sphere ##CC_oo## are formed by adding a single point called ##oo## to ##RR## or ##CC## – the “point at infinity”.
We can then extend the definition of functions like ##f(z) = (az+b)/(cz+d)## to be continuous and well defined on the whole of ##RR_oo## or ##CC_oo##. These Möbius transformations work particularly well on ##C_oo##, where they map circles to circles.
Infinity in Set Theory
The size (Cardinality) of the set of integers is infinite, known as countable infinity. Georg Cantor found that the number of Real numbers is strictly larger than this countable infinity. In set theory there are a whole plethora of infinities of increasing sizes.
Infinity as a number
Can we actually treat infinities as numbers? Yes, but things don’t work as you expect all of the time. For example, we might happily say ##1/oo = 0## and ##1/0 = oo##, but what is the value of ##0 * oo ?##
There are number systems which include infinities and infinitesimals (infinitely small numbers). These provide an intuitive picture of the results of limit processes such as differentiation and can be treated rigorously, but there are quite a few pitfalls to avoid.

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