In a Consistent System, Can a True Sentence Imply a False One?
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From: MoeBlee on 30 Jul 2010 17:48 On Jul 30, 4:34 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jul 30, 3:41 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> 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! > > Are you able to answer that simple question, or not? > > Can you cite for me and for the forum one textbook/source that would > _illuminate_ the meaning and usage of a disprovable formula in the > context of an inconsistent theory? Our lord GAWD! Daryl gave you an adequate quote. The quote doesn't have to stipulate every possible context - inconsistent theory, finitistic theory, inconfinischministic theory! Just PLAINLY, we say a formula it disprovable in a theory iff there is a proof in the theory of the negation of the formula. There's no need to wonder about this kind of theory or that kind of theory. Simply, in ANY theory, a formula is disprovable in it iff there is a proof in the theory of the negation of the formula. I'm DONE! I HOPE! MoeBlee
From: MoeBlee on 30 Jul 2010 17:51 On Jul 30, 4:43 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jul 30, 3:45 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> 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? > > > It STAYS just as it was! > > You are incapable to understand a simple conversation, as usual. > > If it stays "just as it was" why do _you_ need to rename/re-characterize > that to something else that would characterize a consistent theory? What the HELL is your problem? How do you SURVIVE each day in the communication void of yourself? We just happen to have some math lingo that we use now and then to say that the negation of a formula is provable in a theory. Instead of saying "the negation of P is provable in T" we just sometimes say "P is disprovable in T". Often in mathematical conversation we find that there is more than one way to say something. So what? MoeBlee MoeBlee
From: Marshall on 30 Jul 2010 18:12 On Jul 30, 10:39 am, stevendaryl3...(a)yahoo.com (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. Exactly so. Marshall
From: Nam Nguyen on 30 Jul 2010 18:26 MoeBlee wrote: > > What the HELL is your problem? How do you SURVIVE each day in the > communication void of yourself? > > We just happen to have some math lingo that we use now and then to say > that the negation of a formula is provable in a theory. Instead of > saying "the negation of P is provable in T" we just sometimes say "P > is disprovable in T". Often in mathematical conversation we find that > there is more than one way to say something. So what? Shouldn't mathematical reasoning be technically clear where it needs to (such as definitions that would communicate a fact or its negation be the case)? As Daryl pointed out (and indirectly alluded to), "disprovable" would be used to differentiate the 2 cases: "*COMPLETE* consistent theory" (his words) and negation incomplete consistent theory (my words). But both of these _illuminations_ of the technical meaning of the word "disprovable" are _consistent_ theories! You have not yet illuminated why such definition (of "disprovable) would make any technical sense in the case of an inconsistent theory. So don't get angry when you're confronted. For example, if you merely defined a prime as one having only 1 and itself as divisors, then you shouldn't get angry when I or anybody confront you with the question like: Does your "prime" definition make sense in the case of 1? Do you understand this very simple situation, MoeBlee? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 30 Jul 2010 18:30
Marshall wrote: > On Jul 30, 10:39 am, stevendaryl3...(a)yahoo.com (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. > > Exactly so. If you clarified so. Now then, as I asked before, if e.g. I tell you I have a T that has a disprovable formula in it, would you be able to tell if that T is consistent, or not? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt ----------------------------------------------------------- |