I think your summary is wrong, I’m afraid. The sentence could contain two different infinite suffixes—e.g., by interleaving them. (Well, maybe; it still isn’t clear to me what sort of descriptions Thomas is intending to allow.) The problem isn’t that having multiple infinite suffixes is a problem, it’s that (at least for certain rather non-standard notions of what a “sentence” is) there are uncountably many different sentences and if they all have to be described separately you’re dead.
If you allow one description to cover multiple sentences, though, you can cover all those uncountably many with something countably long. Suppose the words of the language are W1, W2, …, Wn and suppose a “sentence” is any string of finitely or countably many of them. (That’s not true for any actual natural language, of course, but this language does have uncountably many “sentences”.) Then you could say: “A sentence consists of the empty string, or: of one of W1,...,Wn followed either by the empty string or by: one of W1,...,Wn followed either by the empty string or by: …...”.
You could also, though this is a further step away from the sort of description I think Thomas wants to allow, do that in finite space. In fact, I already did, earlier in the paragraph above.
I think you’re right. I’m badly overlooking a subtlety because I’m narrowing “describe” down to “is a suffix of.” But you’re right that “describe” can be extended to include a lot of other relationships between parts of the big sentence and little sentences, and you’re also right that this argument doesn’t necessarily apply if you unconstrain “describe” that way. (I haven’t formalized exactly what you can constrain “describe” to mean—only that there are definitions that obviously make our sledgehammer argument break.)
I think “a sentence can be countably infinite” is implicit from the problem description because the problem implies that our “giant block of descriptions” sentence probably has countably infinite size. (it can’t exactly be uncountably infinite)
I think your summary is wrong, I’m afraid. The sentence could contain two different infinite suffixes—e.g., by interleaving them. (Well, maybe; it still isn’t clear to me what sort of descriptions Thomas is intending to allow.) The problem isn’t that having multiple infinite suffixes is a problem, it’s that (at least for certain rather non-standard notions of what a “sentence” is) there are uncountably many different sentences and if they all have to be described separately you’re dead.
If you allow one description to cover multiple sentences, though, you can cover all those uncountably many with something countably long. Suppose the words of the language are W1, W2, …, Wn and suppose a “sentence” is any string of finitely or countably many of them. (That’s not true for any actual natural language, of course, but this language does have uncountably many “sentences”.) Then you could say: “A sentence consists of the empty string, or: of one of W1,...,Wn followed either by the empty string or by: one of W1,...,Wn followed either by the empty string or by: …...”.
You could also, though this is a further step away from the sort of description I think Thomas wants to allow, do that in finite space. In fact, I already did, earlier in the paragraph above.
I think you’re right. I’m badly overlooking a subtlety because I’m narrowing “describe” down to “is a suffix of.” But you’re right that “describe” can be extended to include a lot of other relationships between parts of the big sentence and little sentences, and you’re also right that this argument doesn’t necessarily apply if you unconstrain “describe” that way. (I haven’t formalized exactly what you can constrain “describe” to mean—only that there are definitions that obviously make our sledgehammer argument break.)
I think “a sentence can be countably infinite” is implicit from the problem description because the problem implies that our “giant block of descriptions” sentence probably has countably infinite size. (it can’t exactly be uncountably infinite)