Is there a number so large that we could never calculate it?

Question

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)

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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?

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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!

Answer 1

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.

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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:

• An absolute upper bound on any computer's operational speed is $1/t_{Planck} = \sqrt{\frac{c^5}{Gh}}\ \lessapprox\ 2\cdot 10^{43}\ \tt{bits}\ \tt{per}\ \tt{second}.$
• An absolute upper bound on any computer's storage capacity is
  $Volume_{observable\ universe} /l^3_{Planck}\ \lessapprox\ 9 \cdot 10^{184} \ \tt{bits}.$

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.

Answer 2

You need to define what you mean by calculating a number. I remember finding a website with a program to "calculate a googleplex". What the program really did was output a $1$ followed by $10^{100}$ zeros. Is that calculating a googleplex? Do you really have to multiply $10$ by itself that many times? If you make a rigorous definition, you fall afoul of the Berry paradox. If you don't, the question does not have an answer. Given a way to calculate $N$, I can calculate $N+1$ in the obvious way. When you move from integers to reals, you have the additional result that almost all reals cannot be defined, simply because there are only countably many definitions and there are uncountably many reals.

Answer 3

For calculations, it is the computer memory and how numbers are stored that is the limit.

Simple programs use 32 bit or 64 bit integers, while complex program use string-integers. But even string-integers depend on how much memory is available.

The number $10!^{10!}$ would require about 52 Mb memory to store the number, not to mention the auxiliary numbers that could appear in a recursion step.

The number $10!^{10!}$ has about $54810892$ digits in the number, so yes - such numbers require much memory.

A number like $15!^{15!}$ would require about $33 Tb$ - think about it 33 TERABYTE! Most computers could not even store such a number on the HD!!!

Answer 4

Graham's number.

Graham's number is much larger than many other large numbers such as a googol, googolplex, Skewes' number and Moser's number. Indeed, like the last two of those numbers, the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume. Even power towers of the form $ a ^{ b ^{ c ^{ \cdot ^{ \cdot ^{ \cdot}}}}}$ are insufficient for this purpose, although it can be described by recursive formulas using Knuth's up-arrow notation or equivalent, as was done by Graham. The last 12 digits of Graham's number are ...262464195387.

Answer 5

A necessary condition for being able to calculate a number is that we can specify a computation whose result is that number.

If we accept r.e.s.'s figure that the universe cannot contain more than $10^{185}$ bits, this should mean that $$ \text{the first number whose Kolmogorov complexity is}\ge 10^{185}+1$$ cannot be produced in any deterministic way, in any effective notation.

(It's not just that if we have that number in our hands, we cannot prove that it answers to the above description -- though that is certainly true too. It's that it's impossible to "have that number" in any meaningful way, even without knowing it satisfies that definition).