Can we count past $ \infty $?

Question

Let's say, $ \infty $ is not the biggest number, in fact, there is something bigger, somewhere. We want to try and count past $ \infty $, because it is not really an ordinal one, but a definition of something uncountable or too big to even count.

What numbers are bigger than $ \infty $?

Answer 1

$\infty$ can't be the biggest number, because it isn't a number. It's just a symbol used in limits. Ok, you can consider compactifications of real or complex numbers, but that's certainly not what you had in mind, because both are very far from "counting".

Answer 2

Set ordinalities:

$0,1,2,...,\omega,\omega+1,\omega+2,...,\omega^n,...,\omega^{\omega},...,\omega^{\omega^{\omega^{...}}}=\epsilon_0,...$

One might say it's hugely, vastly, mind-bogglingly big.

Answer 3

The symbol $\infty$, as is currently used, simply means that the value is beyond the range supported by the real numbers. It is not itself a number. For instance, let's say that we only knew single-digit numbers. We might have values, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and $\infty$, to represent a value that is outside the numbers we were familiar with. That's what $\infty$ does. You can't do any sort of arithmetic with it, because it isn't actually specific enough to be a value. It's like when your calculator says "error". You can't actually add two "error" values together, because they aren't really numbers.

However, there are number systems that are beyond the real numbers, and include larger values than are included in the real numbers. Some of these systems include Cantor's transfinite numbers, the surreal numbers, and various hyperreal systems.

Hyperreal numbers are my favorite, as you can basically treat infinities as ordinary algebraic objects. You simply have a symbol to represent some particular infinite value (since we can't actually count to infinity, I can't actually tell you which infinite value it represents), and then treat it algebraically. I usually use $\omega$ for this.

So, in this system, $\omega + 1$ is a distinct number from $\omega$, as is $2\omega$ or $\omega^2$. You can get infinitesimal numbers by dividing by $\omega$. $\frac{1}{\omega}$ is closer to zero than any real number, but is still non-zero. It's really easy to work with, and also simplifies many cases where real numbers have to have a lot of theorems to decide if the value can be contained by the real numbers. In hyperreal numbers, all numbers essentially act as "finite" numbers even if they are infinite, so all of theorems around deciding if a value is finite or not are largely moot.