I was thinking about counting to infinity for some reason or another and I wondered how high one could count. It occurred to me there would be a number you would reach where you could no longer conceive of a higher number due to the physics of the universe.
There are up to 10^80 atoms in the observable universe. If you wanted to try to count in the simplest way possible, you could start counting all the atoms in the universe and eventually you would run out of ways to represent a higher number.
This would be the most basic way of counting, and in our heads right now we can conceive of a much higher number than there are atoms in the universe. In 5 characters, 10^80, we can conceive of this number. That's a lot easier than counting every atom in the universe.
By adding 3 more characters to the end and writing 10^80^80, we have a number with many, many times more atoms than there are in the observable universe. 10^80^80 is a number which would not be able to be expressed by this computer in non exponential notation.
We could take 10 and keep raising it by 80 over and over again until we ran out of atoms in the universe at which point you would reach the highest number available to us. Unless you used a more efficient way of writing numbers. There are ways to express numbers in a more compressed form.
Theoretically there should be some way in which you can arrange atoms the most efficient way possible and express the highest number possibly displayed in the universe.
One might argue, you could just assign arbitrarily large values to each atom and there would be no limit on how high you could count, but this number would have to be comprehensible to some intelligence. Because the universe is closed off to the outside some of the atoms in the universe would have to be used to create this intelligence.
Lets take the simplest version of this to be a key. It might look like this:
. = ..
Which would signify one atom equals the number two. Using this you need at least 6 atoms (not including making the equal sign) for this to be worth the notation as 3 atoms would be equal to the number 6, and 3 atoms were used in the key itself. So if you had only 5 atoms in the universe, the best thing to do would just assume 1 atoms equals 1.
Because you would need 3 atoms for the key,
. = ..
you could use the remaining 2 .. = 4.
Now obviously we have more than 5 atoms in our universe but the principle would be the same. Given any particular number of atoms, there would be a point where using more than a certain number of atoms to represent a certain number per atom would be less efficient. This should be true of any configuration of atoms dictating the counting method which means that there is maximum number in our universe that is knowable.
I found this idea fascinating.