In a Consistent System, Can a True Sentence Imply a False One?
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From: Nam Nguyen on 24 Jul 2010 02:27 Frederick Williams wrote: > Nam Nguyen wrote: >> Nam Nguyen wrote: >>> Frederick Williams wrote: >>>> Nam Nguyen wrote: >>>>> Chris Menzel wrote: >>>>>> On Mon, 19 Jul 2010 07:26:03 -0700 (PDT), George Greene >>>>>> <greeneg(a)email.unc.edu> said: >>>>>>> On Jul 19, 1:13 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >>>>>>>> A system may well be consistent even if some of its axioms are false. >>>>>>> By definition, if the system is consistent, IT HAS A MODEL. >>>>>> True, certainly, for first-order systems, but not by definitiion. >>>>> Otoh, would you think that if it's impossible to know if a system is >>>>> consistent, it could still have a model by whatever process you've >>>>> referred to as "not by definition"? >>>> This "process" is no mystery. See Henkin. >>> It'd be a mystery if it's indeed _impossible_ to know if the >>> underlying formal system is consistent, as my question is really >>> about. >> Let me rephrase my original question to better reflect the problem: >> would you think there could exist a formal system that can carry >> out the basic notions of arithmetic but that it's impossible (even >> in principle) to know its consistency? > > "impossible to know" how? Surely you must know, for example, it's impossible to disprove a formula by rules of inference, because rules of inference can only lead to proof, not to a disproof. That's how! > >> If your answer is yes, would >> it make sense to assume, speculate a model? If the answer is no, why? -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Chris Menzel on 27 Jul 2010 17:14 On Sat, 24 Jul 2010 00:27:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> said: > ... > Surely you must know, for example, it's impossible to disprove a > formula by rules of inference, because rules of inference can only > lead to proof, not to a disproof. That's how! You don't think a proof of ~A is simultaneously a disproof of A?
From: Nam Nguyen on 27 Jul 2010 21:36 Chris Menzel wrote: > On Sat, 24 Jul 2010 00:27:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> > said: >> ... >> Surely you must know, for example, it's impossible to disprove a >> formula by rules of inference, because rules of inference can only >> lead to proof, not to a disproof. That's how! > > You don't think a proof of ~A is simultaneously a disproof of A? > Of course not because in general there cases where there are proofs for _both_ A and ~A. (Naturally, I'm speaking of syntactical proofs using ruling of inference.) -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Chris Menzel on 28 Jul 2010 17:03 On Tue, 27 Jul 2010 19:36:54 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> said: > Chris Menzel wrote: >> On Sat, 24 Jul 2010 00:27:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> >> said: >>> ... >>> Surely you must know, for example, it's impossible to disprove a >>> formula by rules of inference, because rules of inference can only >>> lead to proof, not to a disproof. That's how! >> >> You don't think a proof of ~A is simultaneously a disproof of A? >> > > Of course not because in general there cases where there are proofs > for _both_ A and ~A. Then you simply have an inconsistent system in which everything is both provable and disprovable. So what?
From: Nam Nguyen on 28 Jul 2010 21:31
Chris Menzel wrote: > On Tue, 27 Jul 2010 19:36:54 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> said: >> Chris Menzel wrote: >>> On Sat, 24 Jul 2010 00:27:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> >>> said: >>>> ... >>>> Surely you must know, for example, it's impossible to disprove a >>>> formula by rules of inference, because rules of inference can only >>>> lead to proof, not to a disproof. That's how! >>> You don't think a proof of ~A is simultaneously a disproof of A? >>> >> Of course not because in general there cases where there are proofs >> for _both_ A and ~A. > > Then you simply have an inconsistent system in which everything is both > provable and disprovable. So what? So: a) However much we _believe_ a formal system be consistent, especially one that is "sufficiently complex" (whatever that might mean) there's always a chance our belief might turn out to be simply incorrect! Iow, we shouldn't trust intuition and what's "clearly evident" too much! b) There's a chance that it's impossible (even in principle) to determine if a formula is provable in any extension of a formal system and the underlying extension is still consistent! -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan --------------------------------------------------- |