Is This really a Proof that Standard Math is Inconsistent?
in [Math]
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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
From: spudnik on 21 Jan 2010 13:55 kind of academic, not to actually work a problem in (say) numbertheory, such as Fermat's proof by "infinite descent for n=4 in x^n + y^n = z^n -- the only reuquired non-odd exponent! --les OUevre! http://wlym.com
From: KevinSimonson on 21 Jan 2010 15:14 On Jan 20, 5:17 pm, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote: =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. Thanks, Joshua; this is precisely the answer I was looking for. It definitely gives me something to think about, anyhow. I note, with a big sigh of relief, that standard math may not be inconsistent after all! 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: Marshall on 21 Jan 2010 21:05 On Jan 21, 7:48 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Marshall <marshall.spi...(a)gmail.com> writes: > > On Jan 21, 6:52 am, Don Stockbauer <don.stockba...(a)gmail.com> wrote: > > >> But your discussions aren't totally useless > > > Yours are. > > You don't think his discussions "structure time"? When he said "they" structure time for you I think he meant Sonny and Cher. Marshall
From: Marshall on 21 Jan 2010 23:19
On Jan 21, 12:14 pm, KevinSimonson <kvnsm...(a)hotmail.com> wrote: > On Jan 20, 5:17 pm, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote: > > =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. > > Thanks, Joshua; this is precisely the answer I was looking for. It > definitely gives me something to think about, anyhow. I note, with a > big sigh of relief, that standard math may not be inconsistent after > all! "For every complex problem there is an answer that is clear, simple, and wrong." -- H. L. Mencken You would do well to pay more attention to the replies that have less of the wrong in them, even if it means they also have less of the clear and simple. Look to Hughes, Koskensilta, and Ullrich, for example. Marshall |