In a Consistent System, Can a True Sentence Imply a False One?
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From: Nam Nguyen on 29 Jul 2010 10:05 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Sorry for a typo: I meant to ask: >> >>> Actually, what does it mean for a formula to be "disprovable" in an >>> _inconsistent_ formal system? > > That there is a proof of its negation. So an inconsistent system is also a consistent one then? -- --------------------------------------------------- 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 29 Jul 2010 10:10 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> 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! > > "There's a chance"? We know that in general the problem of determining > whether a formula is provable in a theory is undecidable. Then we don't really have anything to disagree on what I said in b). (Fwiw, I had in mind PA system and the formula pGC when saying b). -- --------------------------------------------------- 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 29 Jul 2010 10:30 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> 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! >> >> "There's a chance"? We know that in general the problem of determining >> whether a formula is provable in a theory is undecidable. > > Then we don't really have anything to disagree on what I said in b). > (Fwiw, I had in mind PA system and the formula pGC when saying b). I do need some coffee: there was a typo. The formula I meant is cGC, not pGC. Sorry. -- --------------------------------------------------- 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 29 Jul 2010 11:10 On Thu, 29 Jul 2010 08:05:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> said: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Sorry for a typo: I meant to ask: >>> >>>> Actually, what does it mean for a formula to be "disprovable" in an >>>> _inconsistent_ formal system? >> >> That there is a proof of its negation. > > So an inconsistent system is also a consistent one then? Obviously not. Since the question has such an obvious answer, I'm guessing that your intention here is to imply that Alan's answer leads to the absurd conclusion that a system can be simultaneously consistent and inconsistent. It doesn't, of course, but why do you think it does?
From: Nam Nguyen on 29 Jul 2010 14:29
Chris Menzel wrote: > On Thu, 29 Jul 2010 08:05:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> said: >> Alan Smaill wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> Sorry for a typo: I meant to ask: >>>> >>>>> Actually, what does it mean for a formula to be "disprovable" in an >>>>> _inconsistent_ formal system? >>> That there is a proof of its negation. >> So an inconsistent system is also a consistent one then? > > Obviously not. Since the question has such an obvious answer, I'm > guessing that your intention here is to imply that Alan's answer leads > to the absurd conclusion that a system can be simultaneously consistent > and inconsistent. It doesn't, of course, but why do you think it does? Because in a T if A is provable but ~A is "disprovable" then T should be consistent. "Provable" means having a proof and "disprovable" means otherwise, in the context of discussing (in)consistency of a system. And Alan was making the definition for an inconsistent theory where all formulas are supposed to be _provable_ (not disprovable). Of course one could rename something to anything, but that would be an odd renaming. -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan --------------------------------------------------- |