In a Consistent System, Can a True Sentence Imply a False One?
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From: Nam Nguyen on 30 Jul 2010 16:41 Daryl McCullough wrote: > Nam Nguyen says... >> Alan Smaill wrote: > >>> The term "unprovable" already exists; >> Right. To be more precise, "unprovable" in technical definition is >> negating "provable". >> >>> "disprovable" is normally used as above -- >>> it does not mean the same thing as "unprovable". >> It actually is, in the context where it's supposed to be used: the >> context of a consistent theory. In such case, the set of disprovable >> formulas and the set of unprovable ones are _identical_ which is >> disjoint from the set of provable formulas. > > No, Godel's theorem shows that the set of disprovable sentences > is *NOT* the same as the set of unprovable sentences. The Godel > sentence G for a consistent theory is unprovable, but it is not > disprovable. > > What you mean is for a *COMPLETE* consistent theory, unprovable > and disprovable are the same. You're right and I had a mistake here, for the consistent theories using the definition of "disprovable" Alan used. In a general consistent theory, the set of unprovable formulas has more than that of disprovable ones. That still does _NOT_ make the definition of "disprovable" he used make sense in the case of an inconsistent theory. For example, if I tell you of a theory T and say there's a "disprovable" formula in T, would you know if T is consistent, or not? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 30 Jul 2010 16:45 Daryl McCullough wrote: > Nam Nguyen says... >> Marshall wrote: >>> On Jul 29, 7:19 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> ... AS' answer wouldn't make much sense in this context of an inconsistent >>>> formal system: all formulas would be _both_ provable and disprovable! >>> Both provable and disprovable! Why, that's hard to imagine. >> Don't tell me but tell Alan that: because that's what his definition >> would render in the case of an inconsistent theory! > > I think Marshall is being sarcastic when he says "that's hard to > imagine". It is *OBVIOUSLY* the case that for an inconsistent theory, > a sentence can be both provable and disprovable. (But it can't be > both provable and unprovable). > > As a matter of fact, we can use the word "inconsistent" to describe > a theory such that some formula is both provable and disprovable in > that theory. What happens to the standard characterization that all a formula and is negation are provable in an inconsistent theory? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 30 Jul 2010 16:49 MoeBlee wrote: > On Jul 30, 12:18 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> To be >> a clown in this context is to be alone > > So does your big red nose honk when you squeeze it? > > MoeBlee The big red nose is his and yours. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: MoeBlee on 30 Jul 2010 17:12 On Jul 30, 3:49 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jul 30, 12:18 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > >> To be > >> a clown in this context is to be alone > > > So does your big red nose honk when you squeeze it? > > > MoeBlee > > The big red nose is his and yours. For a clown, you're awfully unfunny. MoeBlee
From: MoeBlee on 30 Jul 2010 17:14
On Jul 30, 3:41 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Daryl McCullough wrote: > > Nam Nguyen says... > >> Alan Smaill wrote: > > >>> The term "unprovable" already exists; > >> Right. To be more precise, "unprovable" in technical definition is > >> negating "provable". > > >>> "disprovable" is normally used as above -- > >>> it does not mean the same thing as "unprovable". > >> It actually is, in the context where it's supposed to be used: the > >> context of a consistent theory. In such case, the set of disprovable > >> formulas and the set of unprovable ones are _identical_ which is > >> disjoint from the set of provable formulas. > > > No, Godel's theorem shows that the set of disprovable sentences > > is *NOT* the same as the set of unprovable sentences. The Godel > > sentence G for a consistent theory is unprovable, but it is not > > disprovable. > > > What you mean is for a *COMPLETE* consistent theory, unprovable > > and disprovable are the same. > > You're right and I had a mistake here, for the consistent theories > using the definition of "disprovable" Alan used. In a general consistent > theory, the set of unprovable formulas has more than that of disprovable > ones. > > That still does _NOT_ make the definition of "disprovable" he used > make sense in the case of an inconsistent theory. For example, if > I tell you of a theory T and say there's a "disprovable" formula in > T, would you know if T is consistent, or not? Virtually EVERY conversation with you is a method actor's preparation for a scene in the dentist's chair! MoeBlee |