LINGUIST List 2.582

Sat 28 Sep 1991

Disc: Is Language Infinite?

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Directory

1. Stephen P Spackman, Re: 2.573 Is Language Finite?
2. "Larry G. Hutchinson", Re: 2.573 Is Language Finite?
3. Tom Lai, Language is infinite
4. "Michael Kac", Re: 2.573 Is Language Finite?
5. Tom Lai, Is language countable?

Message 1: Re: 2.573 Is Language Finite?

The probable irrelevance of treating langauge as an analogue signal
has already been noted here; but for the sake of the record, it should
also be pointed out that air pressure and time are not real-valued
quantities anyway. There are no real-valued quantities in nature (or
at least, not in the classical, large-cardinality sense of "real",
anyway, and not in the observable-reality sense of "nature"), because
measurement "error" is an inherent quality of physics.
It is most fortunate that language is not large, incidentally, because
otherwise the information content of an utterance would be infinite,
and it would not be possible to process it in finite time (this same
observation applies to numbers: computers can process the constructive
reals, but NOT the classical ones. "Real" numbers in most programming
lanaguages are in fact fixed width floats, all of which are supposedly
rationals (though they actually have the wrong arithmetic for that)).
stephen p spackman Center for Information and Language Studies
stephenestragon.uchicago.edu University of Chicago

Message 2: Re: 2.573 Is Language Finite?

The issue is not how many English sentences could be said. It's how many
English sentences are there? Consider the following (I assume well-known)
example: "John saw --- cows yesterday." Fill in the blank with any natural
number. Isn't the result an English sentence? How many English sentences are
there?

Language is infinite. This is a scientific truth. Jacques Guy is using
the word _infinite_ in a sense that no scientist (at least
mathematicians and computer scientists) should (excuse me for using
this word, but I can't say _will_) use this word in. Jacques Guy's
reckoning of the length of possible utterances and his talking program
show exactly that the no finite maximum exists for the possible length
of an utterance. In scientific jargon, the length of the longest
possible (theoretically speaking) utterance is then at least
_countably_ infinite. In fact, this should be _uncountably_ infinite,
but I am not going into details as it is infinite anyway.
Tom Lai.

Message 4: Re: 2.573 Is Language Finite?

Re Jacques Guy's most recent posting regarding the cardinality of NL's:
The purpose of Langendoen and Postal's *The Vastness of Natural Languages*
is NOT to argue that NL's are infinite -- they assume, along with most o-
ther generatively oriented linguists that this is the case. Their goal
is to argue that NL's are NONDENUMERABLY infinite -- indeed, that they
are maximally so (that is, that the number of sentences in a NL is greater
than any cardinal number).
Anyone interested in this question might find it worthwhile to look at
the paper by Rounds et al. in *Mathematics of Language*, ed. A. Manaster-
Ramer (Benjamins, 1987), particularly the last para. of sec. 4 (p. 354).
Michael Kac

The set of all possible utterances in a language is infinite. In
earlier postings I said the cardinality of this set is uncountable. It
seems that I was wrong. The cardinality of the set of all possible
utterances in a language is countable. This should follow from the
observation that for any positive integer n, the number of different
utterances of length n is finite. This is no big deal. But I would
appreciate confirmation by people out there who are familiar with
these things.
Tom Lai.