In a Consistent System, Can a True Sentence Imply a False One?
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From: Nam Nguyen on 22 Jul 2010 21:54 Chris Menzel wrote: > On Wed, 21 Jul 2010 20:29:03 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> > said: >> 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"? > > It obviously would, if the system is consistent. But the premise of my question is different though: "if it's impossible to know if a system is consistent ..."! > The "process" by which > this is known is a *proof*. It applies to any consistent system, > irrespective of whether its consistency is knowable by us finite, > cognitively limited beings. Our epistemological capacities are simply > and utterly irrelevant to the theorem. > -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Nam Nguyen on 22 Jul 2010 22:09 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. -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Nam Nguyen on 23 Jul 2010 02:30 Nam Nguyen wrote: > Chris Menzel wrote: >> On Wed, 21 Jul 2010 20:29:03 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> >> said: >>> 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"? >> >> It obviously would, if the system is consistent. > > But the premise of my question is different though: "if it's impossible > to know if a system is consistent ..."! > >> The "process" by which >> this is known is a *proof*. But it seems like that's a different kind of proof than the standard first order proof of, e.g., ExAy[~(y e x)] in ZF. >> It applies to any consistent system, >> irrespective of whether its consistency is knowable by us finite, >> cognitively limited beings. But if we can't know the formal system is consistent, what can we really know about its models? >> Our epistemological capacities are simply >> and utterly irrelevant to the theorem. But isn't it true that truths, theorems, and proofs would depend on knowledge, hence on knowledge capacities? -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Nam Nguyen on 23 Jul 2010 09:58 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? 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: Frederick Williams on 23 Jul 2010 11:02
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? > If your answer is yes, would > it make sense to assume, speculate a model? If the answer is no, why? -- I can't go on, I'll go on. |