Is This really a Proof that Standard Math is Inconsistent?
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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: KevinSimonson on 20 Jan 2010 19:00 On Jan 20, 5:51 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: =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. What I have proved is that if T is inconsistent then "P and not P"; "P and not P" is clearly false, so the assertion that T is inconsis- tent must clearly be false as well. That proves that T is consis- tent. And, what is more, it proves that T is consistent _within_ T, since this is a proof by contradiction. Godel proved that in T it's possible to form a statement U that states that it's impossible to prove some other statement V, and does it in such a way that statement V is actually U itself. Either it's possib- le to prove U or not. If it's possible to prove U then U is false, which would make T inconsistent. But I proved that T is _not_ incon- sistent, so it must _not_ be possible to prove U in T. But what I have just stated _is itself_ a proof of U in T. Therefore T is inconsistent. 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: Joshua Cranmer on 20 Jan 2010 19:17 On 01/20/2010 07:00 PM, KevinSimonson wrote: > What I have proved is that if T is inconsistent then "P and not P"; > "P and not P" is clearly false, so the assertion that T is inconsis- > tent must clearly be false as well. That proves that T is consis- > tent. And, what is more, it proves that T is consistent _within_ T, > since this is a proof by contradiction. Not really. Let's examine your logic more closely. Assume T is inconsistent. If it is inconsistent, then one can derive a contradiction. A contradiction is by definition not true, so therefore we have proved our assumption to be false, and T must be consistent. Ultimately, a proof by contradiction assumes that a contradiction cannot be proved in said formal system. You assumed that a contradiction exists in T, which renders invalid a proof by contradiction. -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth
From: James Burns on 20 Jan 2010 19:59 KevinSimonson wrote: [...] > What I have proved is that if T is inconsistent then "P and not P"; > "P and not P" is clearly false, so the assertion that T is inconsis- > tent must clearly be false as well. That proves that T is consis- > tent. And, what is more, it proves that T is consistent _within_ T, > since this is a proof by contradiction. I think you should be more careful with your quantifiers. "T is inconsistent" is equivalent to "There exists at least one statement P, such that P and not P is true." As part of your argument, you assert "P and not P" is clearly false, and it may well be, for particular statements P, but, in your argument that is the negation of "T is inconsistent", that is "For all statements P, P and not P is false." Do you really have a separate argument showing that, for all mathematical statements P, P and not P is false? Then you deserve great praise: that separate argument shows that T is consistent, and there is no need to use the argument assuming that T is inconsistent. However, I suspect that you are just assuming "'P and not P' is clearly false." This is just the assumption that T is consistent. And yes, from T + "T is consistent" you can prove that T is consistent -- this is not within T, though. Jim Burns
From: Jesse F. Hughes on 20 Jan 2010 20:30
Joshua Cranmer <Pidgeot18(a)verizon.invalid> writes: > Ultimately, a proof by contradiction assumes that a contradiction cannot > be proved in said formal system. You assumed that a contradiction exists > in T, which renders invalid a proof by contradiction. This is nonsense. Proofs by contradiction are perfectly valid, whether the theory is inconsistent or not. -- "I am a force of Nature. Time is a friend of mine, and We talk about things, here and there. And sometimes We muse a bit [...] and then We watch them go... in the meantime, Time and I, We play with some of them, at least for a little while." --- JSH and His pal, Time. |