>>I use English, and I suspect that its >>grammar may allow uncountably many sentences. >There are, in the grammars of natural languages, >recursive processes available that make it possible in theory to generate >sentences of infinite length. For example, a sentence S can be imbedded >into another sentence S', in the the following sequence. > 1. John is sick. > 2. I know that John is sick. > 3. I said that I know that John is sick. > 4. I know that I said that I know that John is sick. > 5. I said that I know that I said that I know that John is sick. > 6. I feel sure that I said that I know that I said that I know that > John is sick. >and so on. >>[I have added sci.lang to the news.groups line; ...] >I think they need to see this.We've seen it again and again for several decades, and there's even a book about it: The Vastness of Language, by Paul Postal and Terry Langendoen (who come down in favor of uncountable grammars, within their formal framework, but the question is really moot, for reasons I attempt to explicate below).
There are actually two questions up for discussion here, a linguistic one and a mathematical one, and when they're disentangled, the situation becomes simpler.
First, one shouldn't confuse language with representation of language. Not only in thinking of language as strings of ASCII instead of speech, but also in thinking of it as the formal apparatus of syntactic theory; both of these are artifactual tools for certain representational tasks, and while good engineering always represents something about what it's being engineered for, it shouldn't be confused with the real thing.
In this case, the real thing is real speech, which is full of dimensions (like vowel height, subglottal pressure, and intonation contours) that are engineered using real numbers (or at least double-precision, which is the engineering equivalent of real numbers, and therefore infinite in theory just like infinite-length sentences). You don't have to go to syntax to argue for uncountable variation.
The upshot is that Natural language -- the biological phenomenon that is a specific difference of H. Sapiens -- is of course infinite in this way, but all this means is that it's got continuous variation like any other biological phenomenon.
Are there an infinite number of frogs? Of frogs' croaks?
Grammars of whatever variety are models, engineering models in principle, working models in all too few cases, of the patterns observed by grammarians in their researches of language. Their mathematical characteristics are not usually the reasons why they're adopted, however much fun they may be to play with. In particular, the cardinality of the set of Grammatical Sentences in a Language L in most generative accounts is a function of the method of recursion employed in the model, which allows for tail embeddings of arbitrary length. However, as in most mathematical treatments, it's only the convergent sequences that are of interest, and the first few terms generally settle that. Language rarely goes for five-place accuracy.
So, mathematically (and not linguistically) speaking, of course it's true that there are well-formed "sentences" in this sense (not the linguistic sense) of countably infinite length, and, if you include continuously-valued dimensions of variation, the set of well-formed "sentences" is of uncountably infinite size as well. That's an artifact of the mathematics being employed in the model; it's much simpler to use an unbounded function to estimate a bounded one when you don't know what the boundaries are.
But all this has nothing to do with English or its speakers; the fact is that no English speaker could begin to begin to exhaust the variety of English sentences, and that's close enough for the normal (and not the mathematical) sense of infinite. Unlike language, mathematical infinity is not a topic on which humans have reliable intuitions.
As Jim McCawley once observed, the relation between having a language and a "set of sentences" is not unlike the relation between having a car and a set of trips to the supermarket.