Is This really a Proof that Standard Math is Inconsistent?
in [Math]
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From: spudnik on 20 Jan 2010 00:40 I agree with the above pundits; all you have to do is actually create such a proof, or you can just "work-through" any that were done, such as Fermat's -- http://wlym.com -- the creator of the modern theory of numbers, of which Godel was a rather crude arithmetical usage -- totally elementary, but rather laborious -- I think. in particular, Fermat's "reconstruction" of Euclid's "porisms" is supposed to be exmplary, for a cannonical geometrical proof. > Beware of bugs in the above code; I have only proved it correct, not > tried it. -- Donald E. Knuth thus: bada-ba-BOOM -- hey! > The division sequence repeats the group between A to B, > giving a quotient of 000 111 000 111 000 111 000..., > which at no point has a remainder of 0. So 1001 does > *not* divide 111...111 evenly. thus: first of all, did anyone point out that the Archimedean valuation of "irony" is perhaps a definition of some other (English) word?... I invite others to supply a better word to his putative de-finite-ion! I would, again -- at the risk of contributing to any royalties that AP gets for any one attending to his **** -- like, to refer to Ore's _Number Theory and Its History_ for a de-finite-ive account of Stevin's revolution of _The Decimals_, and the reference to it in Munk's treatise (published by a "vanity press," as he had used during the Great Depression to publish the first "layman's" account of aerodynamics.) [am I recalling correctly, taht this caused the Plutonium One to issue a threat upon my life -- very scarey ?-] > You really should read the articles you quote from. thus: (1) is not self-consistent? how does it materially differ from Gauss's arithmetical series, prime_number + 2n, or what ever? --l'OEuvre! http://wlym.com
From: David C. Ullrich on 20 Jan 2010 07:51 On Tue, 19 Jan 2010 13:05:39 -0800 (PST), KevinSimonson <kvnsmnsn(a)hotmail.com> wrote: >When I was exposed to math in highschool, I was taught that one way to >prove something was to assume the opposite of that something and then >derive conclusions from that opposite. If I ever arrived at a >conclusion that I knew was false, then that would prove the something >I started out with. This was known as proof by contradiction. > >So there is a branch of math, a formal system, I think I can say, >where one can use proof by contradiction to come up with theorems. >And I think it's fairly safe to say that this formal system is very >much in use. > >Isn't it true that the consistency of this formal system comes down to >the assertion that it's not possible to prove a false statement, using >this system's axioms and rules of inference? I think it is true. >Then I will assume the opposite of consistency of this formal system. >I assume that it _is_ possible to prove a false statement, using this >system's axioms and rules of inference. So I _apply_ that proof, and >conclude the false statement. Since my conclusion is false, I have >proven by contradiction that this formal system is consistent. > >But I've proven it's consistent within the formal system itself, which >Kurt Godel proved couldn't be done for a consistent formal system, so >this system must _not_ be consistent. > >Can anybody see the flaw in this argument? Have I _really proved_ >that the math used by the majority of the world is actually >inconsistent? I'm curious to see what everybody else thinks of this. Most of the replies so far point out things like for example the definition of "inconsistent" you're using is not right. This is true, but missing the point - if we insert the correct definition we still get something that appears to be a proof that "the formal system" is inconsistent; we should really be considering why the corrected version is wrong. Ok. Say T is "the formal system". Assume that T is not connsistent. (*) Then T proves "P and not P" for some P. Contradition, so T must be consistent (which then contradicts Godel as you point out). No, (*) is not a contradiction. (*) is not "P and not P", (*) just says that T implies P and not P. Q: But T is "the formal system" used in math, so (*) shows that P and not P is a theorem of standard math! A: No. We haven't shown that T implies P and not P. We've just shown that _if_ T is inconsistent then (*) holds. To deduce what you deduce from this you need to _also_ make the assuumption that T is inconsistent, and that's not part of "stanard math." >Kevin Simonson > >"You'll never get to heaven, or even to LA, >if you don't believe there's a way." >from _Why Not_
From: Don Stockbauer on 20 Jan 2010 08:18 On Jan 19, 3:38 pm, "Mr. Wymore" <wym...(a)ymail.com> wrote: > Proof by contradiction is valid:http://mathworld.wolfram.com/ProofbyContradiction.html > > 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. > > 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 Is it available in paperback?
From: g.resta on 20 Jan 2010 09:42 > > 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 > > Is it available in paperback? If you follow the link above you will see that it is available as paperback and also as used paperback starting from about seven dollars. g.
From: Don Stockbauer on 20 Jan 2010 11:23
On Jan 20, 8:42 am, "g.re...(a)iit.cnr.it" <g.re...(a)iit.cnr.it> wrote: > > > 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 > > > Is it available in paperback? > > If you follow the link above you will see that it is available > as paperback and also as used paperback starting from about seven > dollars. > > g. Thank you so much. I'll have to get a copy of it and expose myself to Hofstadter's mindset. |