brnstnd@kramden.acf.nyu.edu (Dan Bernstein) (04/23/91)
In article <15901@smoke.brl.mil> gwyn@smoke.brl.mil (Doug Gwyn) writes: > There are a finite (although large) number of English words. > The number of combinations of fewer than 13 words from > the finite set of words is also finite (although very large). > The combinations of fewer than 13 English words that express > positive integers form a subset of the set of all possible > combinations of fewer than 13 English words, and thus that > subset is finite. As with all set-theoretic ``paradoxes,'' the flaw is in defining your sets. There exist propositions which do not define sets---in fact, the set of integers not expressible in fewer than 13 words is not defined. If you restrict attention to those propositions which do define sets, the ``paradox'' goes away. This becomes obvious in any formalization of the argument. There is no reason to believe that you can convert any given English proposition into the set satisfying that proposition, just as there is no reason to believe that you can define the set of all sets which do not contain themselves. ---Dan