Raul Miller wrote: On 7/5/07, Terrence Brannon <[EMAIL PROTECTED]> wrote: >> But then again, math is limited by Goedel's Incompleteness >> theorem whereas computers can do more? > > That does not match my understanding, of Goedel's > Incompleteness theorem, nor of the character of computers. > > As I understand the incompleteness theorem: a mathematical > system of sufficient complexity will allow an infinite number > of statements which are not resolvable simply by recourse to > previous axioms (definitions).
I largely agree with Raul, and differ from Terrence. Completeness and consistency are not mutually exclusive. The only mathematical systems that are considered are consistent ones: they may or may not be complete. Boolean algebra is complete: ordinary arithmetic on the natural numbers is not. However this does not mean that every statement on the natural numbers is undecidable (for example, x<:x^2 is decidable and true), only that there are some which are not. Completeness does not have a huge impact on most of mathematics except in areas which directly depend on it, such as the word problem and the halting problem. Computable functions are much smaller than functions from the natural numbers to the natural numbers: there are only countable many of the former, and uncountably many of the latter. As to learning mathematics, I believe that proofs are a post hoc formalization, and that almost all mathematics proceeds from examples. Indeed, theorems without concrete examples are dismissed as "general nonsense". Best wishes, John ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
