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Every natural number is of finite length, but the natural numbers are
infinite. A set does not need an infinite member to have an infinity of
members. Like integers, English sentences may be arbitrarily large, but
each is individually of finite length. If sentences were individually able
to have infinite length, they would be like the real numbers and
uncountably infinite.

On Thu, Jan 17, 2013 at 7:50 PM, Gary Shannon <[log in to unmask]> wrote:

> On Thu, Jan 17, 2013 at 7:19 PM, Roger Mills <[log in to unmask]> wrote:
> > At risk of repeating what others may have said, it isn't that English
> > sentences can be "infinite" in length; it is that there is an _infinite
> > number_ of possible English sentences (or any other natlang).
>
> Without either an infinite lexicon, or the possibility of an
> infinitely long sentence, there cannot be an infinite number of
> possible sentences.
>
> Assume a dictionary of 100,000 words. Just to go to the ridiculous
> extreme and say that any word may follow any other word, there are
> 100,000^2 possible two word sequences. If we allow sentence of N words
> then there are 100,000^N possible sequences. A very large number, but
> large is not infinite. And since only a small portion of all possible
> arrangements are valid sentences, the real number is much smaller.
>
> In general, with a lexicon of X words and a sentence length of N,
> there can be no more than X^N possible sentences. Again, that can be a
> very, very large number, but, again, very, very large is not infinite.
>
> The reason the infinitely long sentence came up in the first place was
> in connection with Cantor's countable vs uncountable infinities. But
> since the number of possible finite-length sentences is finite, that
> doesn't matter.
>
> --gary
>