Is This really a Proof that Standard Math is Inconsistent?
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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
From: William Hughes on 21 Jan 2010 00:14 On Jan 20, 8:00 pm, KevinSimonson <kvnsm...(a)hotmail.com> wrote: > 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. > The problem is that any proofs _within_ T (call such a proof a derivation) must work on statements expressible in T. How do we express T is inconsistent _within_ T? Clearly if T is inconsistent, then there must be a derivation of "P or not P" for some P (and indeed, since proof by contradiction is valid in T, for any P). So There is a derivation of "P or not P" is an expression of "T is inconsistent" _within_ T. However, we cannot go from There is a derivation of X to X without knowing that T is consistent and we would still be arguing outside of T. We cannot derive "P or not P" from There is a derivation of "P or not P" So we cannot prove There is no derivation of "P or not P" _within_ T. (Arguing outside of T we can argue that There is no derivation of "P or not P" is true if T is consistent but this gets us nowhere.) - William Hughes
From: Jesse F. Hughes on 21 Jan 2010 07:54 Andrew Usher <k_over_hbarc(a)yahoo.com> writes: > 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. I have no idea what you're going on about, but my point is simple. 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. -- Jesse F. Hughes 'If you're not making mistakes you're not doing extreme mathematics." -- James S. Harris, extreme mathematician par excellence
From: David C. Ullrich on 21 Jan 2010 08:57 On Wed, 20 Jan 2010 18:48:39 -0800 (PST), Andrew Usher <k_over_hbarc(a)yahoo.com> wrote: >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 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? > is just a statement that logic can never be reduced to >mathematical formulae. > >Andrew Usher
From: William Hughes on 21 Jan 2010 13:04
On Jan 21, 1:14 am, William Hughes <wpihug...(a)hotmail.com> wrote: > On Jan 20, 8:00 pm, KevinSimonson <kvnsm...(a)hotmail.com> wrote: > > > > > 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. > > The problem is that any proofs _within_ > T (call such a proof a derivation) > must work on statements expressible in T. > > How do we express > > T is inconsistent > > _within_ T? > > Clearly if T is inconsistent, then there must be > a derivation of "P or not P" for some P > (and indeed, since proof by contradiction > is valid in T, for any P). So > > There is a derivation of "P or not P" > > is an expression of "T is inconsistent" _within_ T. > > However, we cannot go from > > There is a derivation of X > > to X without knowing that T is consistent > and we would still be arguing outside of T. > > We cannot derive "P or not P" from > > There is a derivation of "P or not P" > > So we cannot prove > > There is no derivation of "P or not P" > > _within_ T. > > (Arguing outside of T we can argue that > > There is no derivation of "P or not P" > > is true if T is consistent but this gets > us nowhere.) > > - William Hughes Ooops. Please read "P and not P" throughout. - William Hughes |