```On Fri, Jan 18, 2013 at 3:21 AM, Mathieu Roy <[log in to unmask]>wrote:

> There are an infinity of finite natural number (and a sentence with
> 1,2,3,etc. words are all possible to make; so there are also an infinity of
> finite sentence). I know this can be hard to understand. But even in
> theory,
> it's impossible to come up with the number of finite possible sentences in
> English (except if you chose a number as the limit of words for a sentence;
> but not if you just say it has to be finite), because than I can come up
> with another sentence a couple of words longer than the longest in the
> list:
> it will still be finite, but I will have add one to the number. In that
> sense, there is no limit to the number of sentences. For any arbitrary huge
> number you tell me, I can find more finite sentences than this number (try
> if you want). This is what is call "infinity", the fact that for any
> arbitrary big (finite) number you give me, I can find more finite sentences
> than that number.
>

This reminds me of something I heard about for mathematics -- that we know
for similar reasons that there are an infinite number of prime numbers (you
can always find a higher one), but no one has ever formally proven it.

But yeah, I think one of the key features of human language is that it is
possible to construct an infinite number of sentences, and (more
importantly) express and infinite number of thoughts.  The only limits here
are physical ones (lifetime of speakers, complexity of the brain, etc) not
limits to the abstract grammatical system.

> That makes me think of a more difficult question but also interesting (IMO)
> which would be to calculate the chance that a random string of X words form
> a sentence (for X equal to, let's say, 15).

That's hard to say.  First, you would have to determine the lexicon you are
generating from (and lexica of languages, I imagine, are arbitrarily large
-- no doubt each individual's lexicon is finite, but it is unrealistic to
expect to be able to catalog all words known by all speakers).  My guess
that generating a random string from any reasonably useful dictionary will
have a very low chance of providing a valid sentence, even allowing
grammatical sentences that are semantically/pragmatically nonsense like
"Colorless green ideas sleep furiously."
```