On Thu, Jul 17, 2008 at 12:04 PM, Günther Greindl < [EMAIL PROTECTED]> wrote:
> > > There are perfectly complete and and consistent axiomatic systems. > (propositional calculus); heck, even the mega-expressive first order > logic (see the completeness theorem). > http://en.wikipedia.org/wiki/Completeness_theorem > > Incompleteness arises when you introduce arithmetic (robinson arithmetic > suffices, presburger arithmetic not; in short: you need addition and > multiplication in your arithmetic -> with this you can construct gödel > numbers, define recursion, and get your (first) incompleteness theorem, > from which second follows easily. > So, if we want computability, we dispense almost all the real numbers, but if we want completeness, we dispense with all the numbers. -- rec --
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