Is This really a Proof that Standard Math is Inconsistent?
in [Math]
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From: KevinSimonson on 19 Jan 2010 16:05 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. 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: Mr. Wymore on 19 Jan 2010 16:38 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
From: KevinSimonson on 19 Jan 2010 17:11 On Jan 19, 2:38 pm, "Mr. Wymore" <wym...(a)ymail.com> wrote: =Godel showed that there was no way to prove ALL of math was true. Isn't that what I just did, prove that "ALL of math was true"? I as- sumed that there was some way to prove something to be true that was actually false, and then arrived at a contradiction. Therefore I con- cluded that everything math proved had to be true, proving the consis- tency of "ALL of math." Was there an error in my proof? If there was, please let me know what it was. Kevin Simonson "You'll never get to heaven, or even to LA, if you don't believe there's a way."
From: Eric Schmidt on 19 Jan 2010 22:33 KevinSimonson wrote: > 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? No. Consistency means that for any statement P, it is not possible to prove both "P" and "not P". -- Eric Schmidt
From: Joshua Cranmer on 19 Jan 2010 23:39
On 01/19/2010 04:05 PM, KevinSimonson wrote: > 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. A consistent system, in short, is one in which a statement of the form "P and not P" cannot be derived. > 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. Try putting your words into more formal logic. It will quickly make no sense. Essentially it boils down to "assume the system is inconsistent; since it is inconsistent, it must 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. YAPBMM: Yet Another "Proof" By Misinterpreting Mathematics. -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth |