What exactly is infinity?

Answer 1

Answer: Infinity isn't exactly anything. When one speaks of infinity, they are referring to an ultimate, i.e. infinitely large is the largest, infinitely small is the smallest, continuing to infinity is never ending, etc. There is no real way to conceptualize infinity since by the mere fact that by thinking of something as being, say, infinitely big, I can immediately think of something bigger by adding one to it, therefore that initial thing wasn't infinitely big to begin with.

The way I conceptualize infinity is in the following way. If there is a finite probability that an event can happen, say the classic monkey writing Shakespeare thought experiment, then given an infinite amount of time, that event will happen, not might happen, but will happen. If that monkey has a 1 in 1010000000000000 chance of replicating Shakespeare, then not only will that monkey do it if you give it an infinite amount of time, it will do it an infinite number of times.

Answer: Since you put the question in math (or accepted the suggested category): In Math, infinity has different meanings in different context.

In set theory, it means, informally, that if you count the elements of a set, you will never reach an end. Formally, infinity can be defined in different ways; for example, an infinite set is one that can be put into a one-to-one correspondence with one of its proper subsets. (It has proper subsets that are "just as big" as the entire set.)

In calculus, it means that an amount has a tendency to go beyond any fixed limit. You might also use the word "unbounded" in this context.

All in all, infinity is not a number. It's a term meaning 'going on and on'.