Is This really a Proof that Standard Math is Inconsistent?
in [Math]
|
From: Don Stockbauer on 24 Jan 2010 14:14 On Jan 21, 4:57 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > "Mr. Wymore" <wym...(a)ymail.com> writes: > > Godel showed that there was no way to prove ALL of math was true. Any > > part of it can be proven by this or other methods. > > Since "ALL of maths is true" is not a mathematical statement at all we > don't needn't invoke Gödel to conclude there can be no question of a > mathematical proof. > > > He also showed that math can be made to contradict itself. Here's a > > cool book that explains it: > >http://www.amazon.com/Godel-Escher-Bach-Eternal-Golden/dp/0465026567 > > I doubt Hofstadter explains how Gödel showed that "maths can be made to > contradict itself", although GEB of course often suggests such misguided > ideas to many otherwise quite sensible people. > > For a sober and readable (semi)popular exposition of Gödel's theorems, > what they say and do not say, what to make of their various purported > implications, uses and abuses, as an antidote to such vague and > wrongheaded musings and reflections as not infrequently inspired by a > half-digested read of GEB, I recommend Torkel Franzén's excellent > _Gödel's Theorem_. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) Actually, Hofstadter does spend a lot of time on Godel's results in GEB, but the book's real value lies in getting people familiarized with the various memes which have formed/will refine the global brain.
From: Don Stockbauer on 24 Jan 2010 15:48 On Jan 24, 1:14 pm, Don Stockbauer <don.stockba...(a)gmail.com> wrote: > On Jan 21, 4:57 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > > "Mr. Wymore" <wym...(a)ymail.com> writes: > > > Godel showed that there was no way to prove ALL of math was true. Any > > > part of it can be proven by this or other methods. > > > Since "ALL of maths is true" is not a mathematical statement at all we > > don't needn't invoke Gödel to conclude there can be no question of a > > mathematical proof. > > > > He also showed that math can be made to contradict itself. Here's a > > > cool book that explains it: > > >http://www.amazon.com/Godel-Escher-Bach-Eternal-Golden/dp/0465026567 > > > I doubt Hofstadter explains how Gödel showed that "maths can be made to > > contradict itself", although GEB of course often suggests such misguided > > ideas to many otherwise quite sensible people. > > > For a sober and readable (semi)popular exposition of Gödel's theorems, > > what they say and do not say, what to make of their various purported > > implications, uses and abuses, as an antidote to such vague and > > wrongheaded musings and reflections as not infrequently inspired by a > > half-digested read of GEB, I recommend Torkel Franzén's excellent > > _Gödel's Theorem_. > > > -- > > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > Actually, Hofstadter does spend a lot of time on Godel's results in > GEB, but the book's real value lies in getting people familiarized > with the various memes which have formed/will refine the global > brain. If you want to solve a problem, build a better tool (or watch while yourself and other systems self-organize into a bigger tool).
From: Andrew Usher on 24 Jan 2010 19:25 On Jan 21, 6:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Rules of inference are valid or not independently of the theory in > which they are used. The definition of validity does not refer to the > theory. Well, not my definition. If you say so, then your logic is just meaningless symbol manipulation. Andrew Usher
From: Andrew Usher on 24 Jan 2010 19:26 On Jan 21, 7:57 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > >Only in a vacuous sense. Mathematicians do assume 'P xor not P' > >because it is true, that is, true in real, informal logic. The fact > >that Goedel's theorem shows that it is not always so in any formal > >system > > For heaven's sake, where did you get the idea that Godel's > theorem says that "P xoe not P" is not always so in any > formal system? It shows that it isn't always provable, which in the context of rigorous proof means the same thing. Andrew Usher
From: Daryl McCullough on 24 Jan 2010 20:11
Andrew Usher says... > >On Jan 21, 6:54=A0am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Rules of inference are valid or not independently of the theory in >> which they are used. The definition of validity does not refer to the >> theory. > >Well, not my definition. If you say so, then your logic is just >meaningless symbol manipulation. Well, in a sense, that's what logic is all about; the study of arguments that are valid because of their form. The point of a logically valid argument is that its validity should be checkable *without* any knowledge of the meanings of the function symbols, predicate symbols, or the domain of discourse. -- Daryl McCullough Ithaca, NY |