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

Solution 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.

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.

Solution 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.

Solution 3:

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).