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On Jan 17, 2013 10:50 PM, "Gary Shannon" <[log in to unmask]> wrote:
>
> Without either an infinite lexicon, or the possibility of an
> infinitely long sentence, there cannot be an infinite number of
> possible sentences..

No, this is incorrect.  You are confusing "of finite length" with "have an
upper limit".  Take the whole numbers as an example.  There are infinitely
many, however, none is of infinite length. It is simply that you cannot
pick a "maximum".  The same is true of sentences (depending of course on
one's definition of sentence). If you say "a sentence may not be longer
than N words", then there cannot be infinitely many sentences, however if
you say "sentences must be of a finite length" then there can.

Someone has previously offered a proof of this, but I'm on my phone which
makes it difficult to look back, so I will restate it.  Imagine you have a
finite list of all sentences.  You can construct a new sentence by adding
an embedded clause to the longest sentence on your list.  Therefore the
list did not contain all sentences.

@Alex, there are an infinite number of integers, and integers cannot be on
infinite length.  The same argument above applies.  The important
distinction is that "has an upper bound" and "are of finite length" are
different statements.

-Daniel