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There are other arguments based on the hubble volume of the universe and how many states it can contain (by calculating the number of particles that can be crammed into that space, that calculation, discussed in a scientific american article was 10^118).

In any case, that means if you pick a sufficiently large number, then you can say that all the number from 1 to that number can not be represented or expressed in this universe. And if there is no way to even theoretically express a range of numbers, how can you say they exist?