Is This really a Proof that Standard Math is Inconsistent?
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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.
From: Andrew Usher on 20 Jan 2010 21:48
On Jan 20, 7:30 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Joshua Cranmer <Pidgeo...(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. 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 is just a statement that logic can never be reduced to mathematical formulae. Andrew Usher |