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- Jacques Guy, Is language infinite?
- Tom Lai, Why talk about infinite lang
- John Coleman, RE: 2.595 Is Language Infinite?

I have received this message which encapsulates the gist of this discussion: "I can only assume that you believe there is a finite number of numbers, by precisely the same argument as you apply to natural languages. [...] If you believe there is an infinite number of numbers, then you contradict yourself and should rethink what you believe about natural language." The assumption is right. Yes, right; LITERALLY right. "Literally" is the key: I do hold that the "number of numbers", that is, the number of PHYSICALLY enumerable SIGNS representing numbers, is finite. I also hold that the number of positive integers is infinite. Ditto for the number of points on a line, or in a plane, or in a space, only more "infinite" than the former. Contradiction? No. Integers, geometric points have no physical existence; linguistic signs have. There again we meet the fundamental distinction between signifiant and signifie', sign and referent, and between reality and abstract models of it. Should I ramble on about what lurks behind "numbers" and "counting"? I'd rather not. Show time, instead. ------------------------ ARTHUR: This new learning amazes me, Sir Bedevere. Explain again how there are infinitely many ways in which Joseph could have descended from Abraham. BEDEVERE: Of course, my Liege. Matthew has written that Abraham begat Isaac, who begat Jacob, who begat Juda, and so on, who begat Joseph. ARTHUR: Yes. BEDEVERE: Behold then, my Liege, these generative rules: <Genealogy of Joseph, from Abraham> ::= Abraham <more Hebrews> Joseph <more Hebrews> ::= <one Hebrew>| <one Hebrew><more Hebrews> They are truly wondrous, my Liege, for they account for all the possible filiations through which Joseph could have descended from Abraham, including that reported by Matthew: just replace <one Hebrew> with a Hebrew and have him beget. ARTHUR: Uh, yeah, I can see that. Randy lot those Hebrews, eh? BEDEVERE: And those alternative genealogies, my Liege, are infinite in number because <more Hebrews> is recursive. ---------------------------- As you can see, Sir Bedevere's conditions for membership of the set of alternative genealogies include a recursive definition, which allows him to conclude that their number is infinite. Reinterpret "Abraham" and "Joseph" as the boundaries of a sentence, and replace <one Hebrew> not with a Hebrew patriarch but with a phoneme of whatever language you please. Sir Bedevere's rules now generate all the possible sentences of that language, and the impossible ones too. Are there really infinitely many possible alternative ways in which Joseph could have descended from Abraham? Yes, if there are infinitely many sentences in any natural language. And vice versa. For the rules are one and the same, and the cardinality of their elements alike: there was a finite number of generations from Abraham to Joseph, as there is a finite number of phonemes in any sentence; there was only a finite number of Hebrews draftable into stud duty at any time, as there is only a finite number of phonemes you can choose from to fill any particular position in a sentence. >From here on, the choice is yours. But note that Sir Bedevere does not test what his model predicts against what is physically possible, let alone what has been physically observed. He is in good company, with Hegel and Descartes, for whom truth was derivable through reason alone without reference to the physical world. Does that settle it?Mail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue

When we look at a language in its actual state, it is finite. As such it has all the limitations of finite entities. There are, for example, ideas that do not have a word in the lexicon to express it with because the lexicon is finite. But, if we consider the possibility of an open lexicon taking in new words, the infinite language, in its abstract aspect, has the potential to express anything hitherto not encountered. So, it is not meaningless to talk about the properties of of a language that come with its infinite character. I understand that the question of whether sentences can be of infinite length is a thorny issue. I think I should not say anything definite about it. Tom Lai.Mail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue

macrakisosf.org asks > 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 [Langendoen > and Postal] get anything bigger? They allow infinite-lenth sentences, built by closing a finite set of finite-length sentences under conjunction. They they introduce a Cantorian diagonalization to show the existence of new sentences not in the original set, and thus claim to show that the size of an NL is transfinite. Given that the argument they present is essential no different from Cantor's, it seems to me that if you accept the "existence" in some sense of transfinite numbers, you can't also deny the existence of sentence-like objects and transfinite-sized NL-like objects of the sort Langendoen and Postal are talking about. The philosophical problems only arise when we consider whether the NL-like objects of transfinite size are "real languages" in the more familiar sense. But this is the bag of worms, not L&P's mathematical argument, it seems to me. --- John ColemanMail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue