Note that I edited this post significantly to make it more clear (as clear as I think I could possibly make it). First, let me mention what I am NOT asking:
- I am NOT asking for the largest number we could calculate.
- I am NOT asking if there is a largest number (if there were, just add one and it would be bigger, so obviously this is false)
- I am NOT asking for something like "divide by zero" or for limits ("infinity" isn't an answer, that isn't a number)
I have been thinking a lot about numbers and I put a couple numbers into Wolfram Alpha just to see what happens. It can handle insanely huge numbers, but it fails as soon as you put in $10! ^ {10!}$. After thinking for a few days, I feel that there must be a largest number that we could possibly ever calculate or define. In fact, I think there are an infinite number of numbers greater than anything we could ever calculate (I am saying larger numbers exist, but we cannot do anything with them, except maybe prove there are numbers so large we can't calculate them).
I mean a number or an equation that results in an exact number (such as $x!^{x!^{x!^{x!^{x!^{…}}}}}$, where $x$ is an exact number that can be written out) I think there is a limit here. There is a largest number possible because there are only so many atoms in the observable universe. But how would one prove this (that there are numbers so large we couldn't define them)? Is there any information on this?
Interestingly, of the irrational numbers we use, they always have an alternative way to write them such that the answer is exact. One can simply write the infinitely long irrational number that is the square root of two as $\sqrt 2$. Two symbols and you define the infinitely long number that is the square root of $2$. One can use the formulas for finding $\pi$. But aren't there numbers that we couldn't even do that for? Aren't there numbers that we couldn't write in some form, such as how we write $\sqrt 2$? Such that the number is exact? I think these numbers must exist, but we can't do anything with them. But I have no idea how one could prove that or if there is any information regarding these numbers. So my question is: are there numbers that are just too large (or too precise) that we can't write them down in any way at all that expresses their exact value?
UPDATE:
I accidentally came across something that really helps with my question. I found this article here, and I saw point #6:
Unknowable Thing: There are numbers that can’t be computed.
This is another mind bender proved by Alan Turing.
That is exactly what I'm talking about! I am going to do some research on Alan Turing!
Best Answer
It can be shown that in the context of ordinary mathematics (say ZFC) there are infinitely many well-specified positive integers whose numerical representations cannot be proved. E.g., for every $n \ge 10\uparrow\uparrow 10$, the Busy Beaver number $\Sigma(n)$ is well-defined and has some decimal representation $d_1d_2...d_k$, but there exists no proof that $\Sigma(n) = d_1d_2...d_k$. It isn't that the proof or the digit string is merely infeasible due to physical resource limitations; rather, such a proof is a logical impossibility.
Here are a few relevant online sources:
The Wikipedia article on Σ, complexity and unprovability.
This answer to the math.SE question How can Busy beaver($10 \uparrow \uparrow 10$) have no provable upper bound?.
Pascal Michel's webpage on Busy beavers and unprovability.
NB: In connection with the computability of numbers, note that an uncomputable number cannot be an integer (because each integer has a purely finite representation, unlike the situation for real numbers). That's why the "computable-but-unprovable" results mentioned above seem especially poignant, since they apply specifically to positive integers, without complicating the situation with infinite objects such as the digital representations of uncomputable real numbers.
In a completely different (and much more mundane) sense, a digital representation of a positive integer can be "too big to calculate" for reasons of physical infeasibility implied by the assumed laws of physics:
$Volume_{observable\ universe} /l^3_{Planck}\ \lessapprox\ 9 \cdot 10^{184} \ \tt{bits}.$
See the Wikipedia article on Physical limits to computation, and also the absolute bounds mentioned in the external weblink provided in the article on Bremermann's limit.