I thought Gurkenglas' solution was a lovely discrete math sledgehammer approach. There's a lot of subtly different problems that Thomas could have meant and I think Gurkenglas' approach would probably be enough to tackle most of them.
(Attempting to summarize his proof: Some English sentences, like the one this problem is asking you to dig around in, are countably infinite in length. If some English sentences are countably infinite in length, and any two of them have different infinite suffixes, then there's no way the text of this sentence contains both of them.)
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, ... (read more)
One of the problems was that Thomas spoke about a "natural language" that we all know what it looks like. And a natural language does not have infinite-length sentences.
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