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This is a serious question I started thinking about this evening and it turned into quite an existential rabbit hole.
The question is this: what is the smallest positive whole number which cannot be unambiguously expressed in this universe?
Naturally the point isn't to answer the question per se, so it would help to consider simpler hypothetical universes, the nature of language, etcetera; maybe such a number doesn't even meaningfully exist?
As I see it, it there is no lower bound on the number without placing restrictions on the "expressions" involved. Specifically, it would seem that any such expression would have to be a map from the set of possible universes to the positive whole numbers. Take any such map, call it θ. Then let φ = θ + 1 so that φ(U) = θ(U) + 1 for some universe U. Then φ(U) > θ(U).
So it would be dependent on restrictions placed on the set of allowable mappings. One obvious way to do this would be to index the set of possible states that the universe could be in. If there are a finite amount of states, then the amount of states is the largest such number. If there are denumerably infinite states, then there would be no largest integer. If there are more states than that, this technique doesn't make sense.
Is it true, though, that a universe with n possible binary states (which seems like the best interpretation of 'universe' in this context) can express a maximum of 2^n numbers (i.e. the maximum binary number with n digits)?
Numbers do not necessarily need to be expressed in positional notation to be unambiguous. For example, expressions like 2^(2^n), or p(n) where p(k) is the kth prime are unambiguous and refer to numbers greater than 2^n, which seems to greatly expand the range of expressible numbers.
Right, absolutely. The reason I went with indexing is because there needs to be some limit placed on the process as I showed in the first paragraph of my first post. Otherwise, the answer to your question is trivially, "No there is no smallest such number." Let's take your example, p(k).
Suppose that you present me with a means of expressing a number with a universe, α; a universe, U; and a number, m, and claim m to be the minimum ineffable number. Then I can just return p.α (. indicates composition) and have a larger representation as we have that p(m) > m.
So you need to specify a particular mapping, in which case the problem becomes boring, or a family of mappings, in which case it may get interesting.