By definition, infinity is a behavior not a number. However, in schools we see equalities like
so we should treat this as a number. But what about expressions like
∞-∞ and ∞/∞
The answer got to us back when we were at school. Neither of these expressions is well-defined.
Let’s get back to our first example, if you add 2 extremely large numbers together, you get another extremely large number. Same thing if you are multiplying.
When it comes to subtracting or dividing them we don’t know the outcome. This is why subtraction in infinity isn’t defined.
Different kinds of infinity
Infinity describes many different concepts. The natural numbers can’t be bounded from above. Any number x that you choose from the natural set, you can find x+1 ( ∞+1 doesn’t exist) that is larger. However, if you have an infinite amount of time, you can name all of the positive integer numbers. Whereas, if you try to count all of the real numbers, assuming that you have an infinite amount of time, there will always be a number that you have missed.
The natural set kind of infinity is called “countably infinite” and the infinity over the real number set is called “uncountably infinite”.
So when Buzz lightyear says “to infinity and beyond (∞+1)”, he is not totally right!