Sure, but this is missing the point. There’s also a finite number of C programs that can actually exist in the physical universe, but that’s an arbitrary limit,

Sure, but this is missing the point. There’s also a finite number of C programs that can actually exist in the physical universe, but that’s an arbitrary limit, not part of the definition of C. Similarly, there’s no non-arbitrary limit on, say, the length of an English sentence. All of this ‘debate’ about the infinity of language just reduces to a misunderstanding of what people are saying, as far as I can see. No-one thinks that more than a finite number of English sentences will ever be uttered; conversely, no-one thinks that we will ever discover such a thing as a complete list of all English sentences (since trivially the conjunction of all these sentences would be a new sentence not on the original list). Note that if you view English as a regular tree language, your point about chess also applies to English. You don’t need to remember all of the preceding state, just the congruence class. No-one is saying that English is “infinitely complex”. The grammar of English is finite, but there is no limit on the number of sentences that it can assign a structure to. Of course you can encode intensions using numbers in the broad sense that you can encode pretty much anything using numbers.