In a Consistent System, Can a True Sentence Imply a False One?
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From: Jim Burns on 20 Jul 2010 16:32 Daryl McCullough wrote: > Jim Burns says... > >> Charlie-Boo wrote: >>> In a consistent system, can a true sentence imply a false one? >> >> The most straightforward answer to your question is "No". >> > > Charlie said *consistent* system, not *sound* system. A consistent > system only guarantees that you can't derive a contradiction. There > is no requirement that you can't derive false conclusions. Thank you for correcting my error. Jim Burns
From: Chris Menzel on 21 Jul 2010 09:04 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.
From: Nam Nguyen on 21 Jul 2010 22:29 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"? -- --------------------------------------------------- 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 22 Jul 2010 10:15 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. 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.
From: Frederick Williams on 22 Jul 2010 14:36
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. -- I can't go on, I'll go on. |