Re: Infinite number of words?

In article
sable <zxcv_890@xxxxxxxxxxx> wrote:

I'm reading a book _Intro to Computer Theory_ by Daniel Cohen. In chap
2, he talks about defining languages, and he uses English as an
initial example. The English alphabet consists of the letters a
through z plus the hyphen and apostrophe. He says (p. 8) "We can now
specify which strings of these letters are valid words in our language
by listing them all, as is done in a dictionary. It is a long list,
but a finite list, and it makes a perfectly good definition of the
language." Initially I accepted this, until he started talking about
formally defining the language of English *sentences*. He says (p. 9),
"In order to specify which strings of elements from [the dictionary of
English words plus the space plus the punctuation marks] produce valid
words in the language ENGLISH-SENTENCES, we must rely on the
grammatical rules of English. This is because we could never produce a
complete list of all possible words in this language... Theoretically,
there are infinitely many different words in the language ENGLISH-
SENTENCES. For example,

I ate one apple.
I ate two apples.
I ate three apples.
" (end quote)

Wait a minute! He's implying that there are infinitely many English
sentences because of the infinite number of numeral words! Then why
couldn't you apply that same argument to the English dictionary and
say that you could never produce a complete list of all dictionary
words? My point is that if it's indeed true that there are infinitely
many English sentences, he used a completely inappropriate example to
illustrate it (or am I wrong about that?) Wouldn't it have been more
correct for him to argue that there are infinitely many sentences
because there is no limit to the number of words/clauses that can be
appended to a sentence? Since I'm pretty new to this, I'd appreciate
any feedback on whether my criticism is fair.
Thank you.

I agree with you. In practice we use only finitely many number
words (to be found within a dictionary). Perhaps a better example of
infinitely many sentences might be:
Boring music goes on.
Boring music goes on and on.
Boring music goes on and on and on.

Ken Pledger.

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