franka@mmintl.UUCP (Frank Adams) (10/09/87)
[I am moving this discussion to philosophy.tech, since it seems to have become a philosophy of mathematics issue as much as anything else.] In article <171@yetti.UUCP> peter@yetti.UUCP (Peter Roosen-Runge) writes: |However, if you do feel convinced that infinite sets actually exist and |that natural languages as sets are or can be infinite, why stop |at the countable? It turns out that the arguments for stopping at that |cardinality aren't as strong as people used to think -- a beautiful hatchet |job on the countability restriction has been provided by |Langoeden and Postal in THE VASTNESS OF NATURAL LANGUAGE, Blackwell: 1984 |They argue that languages are far larger than countable and flit loftily up |into the rarified stratosphere of transfinite cardinals and mega classes. | |The authors state that one does not have to be a platonic realist to accept |their arguments but it sure helps! At the very least, the book is extremely |helpful in making clear what is at stake in allowing non-constructive |descriptions of infinite classes. It is a hazardous enterprize to argue against an argument which has not been stated, but I am going to attempt it anyway. I am assuming that Messers. Langoeden and Postal are arguing that a sentence like: "Consider a real number x." has uncountably many meanings, since x can be any of uncountably many values. I think this is wrong; it has only one meaning, about an incompletely specified value x. Allowing non-constructive descriptions of infinite classes does not commit us to believing that we can refer to particular non-constructive objects. -- Frank Adams ihnp4!philabs!pwa-b!mmintl!franka Ashton-Tate 52 Oakland Ave North E. Hartford, CT 06108
franka@mmintl.UUCP (Frank Adams) (10/10/87)
[Followups are directed to sci.philosophy.tech] In article <905@sjuvax.UUCP> tmoody@sjuvax.UUCP (T. Moody) writes: |As far as I know, no general effective method exists for determining |whether any given string is or is not a sentence of a [natural] language. | ... If it could be proved that no such effective method exists, then the |set of sentences in a language would be undecidable. How does this affect |its cardinality? Depends on your mathematical philosophy. For a classicist of any type, it doesn't affect it in the slightest. For an intuitionist, you first have to ask what "cardinality" means. -- Frank Adams ihnp4!philabs!pwa-b!mmintl!franka Ashton-Tate 52 Oakland Ave North E. Hartford, CT 06some