What are the largest numbers in physics?

Question

In mathematics there are some big numbers such as the busy beaver numbers which are large but have a short description. I was wondering if we allow physical definitions of numbers we can make some bigger ones. So how big is this number (very approximately), assuming we live in a spatially infinite universe, how far must we travel on average before hitting on a copy of the earth that no human can discern from our own? (Divided by the average distance of an electron to atomic core to make it dimensionless). Can we construct a number larger then BB(BB(99)) by allowing reference to physical objects in the definition? Here is another example, how long in time until universe reaches thermodynamic equilibrium? How many possible quantum states is there for the universe?

No numbers which can be defined strictly mathematically is allowed. No numbers which is basically a function taking as input a large number from mathematics is allowed. Only dimensionless numbers count.

Alternate phrasing:

Let f(n) be the largest number definable using n characters including spaces, in PA or some formal system.

Then we can define g(n) to be the largest number definable using n characters including spaces, where the english language and any reference to a physical quantity such as number of particles in the universe is also allowed.

My question is, is g(n)=f(n)?

Comments

Voting to close. The question is vague to the point of being meaningless. For example, BB(BB($N_{Avogadro}$))) is vastly larger than  BB(BB(99)). And the number of possible quantum states of a hydrogen atom is already infinite. 

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No. You can define that number strictly mathematically

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Every number with a physical meaning can be defined strictly mathematically in terms of the fundamental constants. 

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I think there could be an interesting question here. The rule is you're only allowed to apply simple algebraic functions to physically defined (ie. measurable) numbers.

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@RyanThorngren: Unless the question is made much more precise, it is meaningless: The number of quantum states of a given metal bar that are distinguishable at a givel accuracy $\delta$ is huge and goes to infinity as $\delta\to 0$, hence gives arbitrarily large numbers with a physical meaning. There are plenty of similar numbers, e.g.,  the number of possilbe classical states of a flask of fluid hydrogen at a fixed time whose molecular positions and momenta are fixed to a givn precision $\delta$.

Note that these numbers are directly in the spirit of the question as currently phrased!

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@Arnold You havent given a number, but a function, which is computable from the postulates of quantum mechanics, BB(n) however grows faster then any computable function and thus faster then yours. It could be that we could define a number physically, but that it was too large to admit a mathematical description even if we used the best compression and all bits physically available to us. BB(n) is defined in terms of turing machines, maybe if one defines a similar function for quantum computers one could get one faster?

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This question is mostly for busy beavers, not for Physics Overflow users. We have here some overflow, but not that much.

Answer 1

The question is physically meaningless and instead illustrates a natural language paradox.

Let N be the smallest number not definable using 212 characters including spaces, where the English language and any reference to a physical quantity such as number of particles in the universe is also allowed.

Since this yields a definition of N using 212 characters, we have found a contradiction. Therefore there is no such smallest number. This means that every natural number is definable using 212 character, according to the specification of the question.

Therefore g(212) is not defined. 

Even worse, since there are only finitely many sentences with at most 212 characters, the number of numbers definable in the specified way is finite. Therefore the argument contradicts the well-known theorem that there are infinitely many natural numbers. 

Thus we have found a contradiction in the logical system consisting of the Peano axioms for the natural numbers and the English language, and haven't even used that using physical quantities is allowed.