One of the mistakes students often make is assuming (justifiably) that infinity is a number. It’s a natural assumption. Numbers are the things we talk about in math, after all, and we treat infinity as a number in some ways. We say things like, “You have an infinite number of toothbrushes.” (It’s clearly none of my business what you are doing with that many toothbrushes.)
What does infinity mean? Let’s go back to the playground for a minute. Double dares turn into triple dares, which eventually turns into cries of, “I infinity dare you!” It is the numerical trump card. Something larger than any other number. In Division by Zero I made the assertion that infinity is either a number or it’s not. That’s how properties work. If it is both a number and not a number that would be a contradiction, which we cannot accept.
So, let’s assume that infinity is a real number. If we couple this with our schoolyard knowledge, then we must say it is the largest real number. (If it weren’t the largest number, then we would just take some larger number to be infinity instead — just like our childish predecessors.)
Since we are assuming infinity is a real number, then ordinary operations like addition are defined for it. This means that infinity + 1 is also a real number. Adding one to a number makes it larger. So infinity + 1 larger than infinity. This is a contradiction. So, we must conclude that our assumption was wrong. Infinity is not a real number.
So, if it isn’t a number, what is infinity? It is a concept — the notion of going on forever. I cringe with Woody every time I hear Buzz’s catchphrase.