Is Language Bound By The Incompleteness Theorem?

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davidbackslashse
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Re: Is Language Bound By The Incompleteness Theorem?

Postby davidbackslashse » Thu Nov 05, 2009 2:25 am UTC

bebboe wrote:
davidbackslashse wrote: Natural languages (the kind we speak) are recursive and are capable of generating sentences of infinite length (in practical terms, though, we're really only capable of processing up to about three embedded clauses). Formal languages, as in the quote, are defined over the set of strings in full that make them up.

This is a common misunderstanding. There is no such thing as a sentence of infinite length, at least not in natural languages. The length of sentences in natural languages is unbounded, but not infinite. There is no upper bound to the length of sentences, that is, given any sentence, you can always construct a sentence that is longer than it, but this is NOT the same as saying that there are sentences of infinite length (though it follows from this that the cardinality of the set of English sentences is infinite). Not that this really matters in this discussion, but since even some of the most educated people I know tend to claim that Chomsky claimed that there were sentences of infinite length, I felt like clearing this up.


You're right, I phrased that badly. Thanks!

Supergrunch
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Supergrunch » Sun Nov 08, 2009 10:30 pm UTC

davidbackslashse wrote:
Noam Chomsky wrote:From now on I will consider a language to be a set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements.


isn't about natural languages; it's about formal languages. Natural languages (the kind we speak) are recursive and are capable of generating sentences of infinite length (in practical terms, though, we're really only capable of processing up to about three embedded clauses). Formal languages, as in the quote, are defined over the set of strings in full that make them up.

http://en.wikipedia.org/wiki/Chomsky_hierarchy

That quotation is taken from Syntactic Structures (opening of section 2, p13 of the 2nd edition), and unambiguously applies both to natural languages and to many formal languages. Here's some more context:
Noam Chomsky wrote:From now on I will consider a language to be a set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements. All natural languages in their spoken or written form are languages in this sense, since each natural language has a finite number of phonemes (or letters in its alphabet) and each sentence is representable as a finite sequence of these phonemes (or letters), though there are infinitely many sentences. Similarly the set of 'sentences' of some formalized system of mathematics can be considered a language.

(emphasis in the original)

This is from Chomsky's first book to be published (1957), so doesn't necessarily agree with his current ideas, though this quotation doesn't really conflict with them greatly as far as I can tell.


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