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- , 2.582 Is Linguist Infinite?
- "Michael Kac", Re: 2.582 Is Linguist Infinite?
- Pete Humphrey, Is Language Infinite
- , 2.582 Is Linguist Infinite?

Michael Kac writes: 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). As one of the authors of the article, I followed the instructions and read the last para. of sec. 4. It turns out that we pointed out that it is indeed reasonable to take languages as finite. However, this only makes sense if we treat them not as fixed finite sets but as families of finite sets with no fixed bound on size. As far as I know, Yuri Gurevich at U. of Michigan and some other people have developed a formal model of this sort to serve as a model of the behavior of computers. But I should caution that not much rides on the distinction between infinite sets and such families of finite sets, although there are some arguments for the latter view as more realistic (both for computers and for human beings). I would also add that I am surprised by the vehemence with which several correspondence assert as fact or as scientific truth something that I would regard as a matter for indirect argument at best and perhaps only of mathematical convenience. The same applies, of course, the Langendoen/Postal work: I still cannot believe that Terry and Paul could seriously claim that the issue of whether NL sentences are merely of unbounded length or of infinite length (and whether the collection of English sentences was countable or not) a factual question. But by the same token I cannot understand the self-righteous dismissal of their proposals by those who somehow possess the certitude (that I so notably lack) that NL sentences are only of finite length. The same applies to the question of whether there are any truly analog phenomena in nature. I do not feel at all certain that nature is ultimately digital as some seem to. Finally, in response to Tom Lai's query, the following is mathematically correct (this we can be certain of): If we assume that there is no upper bound on the length of English sentences, then there is an infinite (but only countable) set of these, since they can then be placed in one-to-one correspondence with the natural numbers. If we assume that English sentences can be of actually infinite length, then (depending on what else we assume), they may form either a countable or an uncountable set (it is crucial that we assume something more, because you can have a finite (even a singleton) set of infinite-length sentences).Mail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue

Footnote to my reference to Rounds et al.: The question addressed in the relevant parts of the paper (one of which I explicitly mentioned) is the question of whether NL's are infinite, not that of whether they're denumerable. Post facto, I found my own wording confusing. MKMail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue

It seems that any language, in written form, would be countable. The n symbols of the writing system can be taken as the digits of a base-n number and every sentence in the language will be a finite string of these digits. This gives a simple one-to-one correspondence to the natural numbers. - Pete HumphreyMail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue

Kac says: Langendoen and Postal's *The Vastness of Natural Languages* ... 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). Assuming finite-length sentences and finite numbers of combining elements (whether phonemes or something else), set theory won't get you anywhere beyond countable infinity. How do they get anything bigger? By the way, although mathematics talks <<about>> vast cardinals, only a countable number of mathematical objects can ever be individually named. Is this the paradox they play on, perhaps? -sMail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue