In a Consistent System, Can a True Sentence Imply a False One?
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From: Nam Nguyen on 30 Jul 2010 17:28 MoeBlee wrote: > 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. LOL. And you're so funny MoeBlee! Didn't I say you're a clown. I actually did! -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 30 Jul 2010 17:34 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? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: MoeBlee on 30 Jul 2010 17:37 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! Try to say these words, Nam: Definition: In a theory T, a formula P is provable iff there is a proof in T of P. Definition: In a theory T, a formula P is disprovable iff there is a proof in T of ~P. Definition: In a theory T, a formula P is unprovable iff there is no proof in T of P. Definition: A theory T is consistent iff it is not the case that there are proofs in T of both a formula P and of ~P. Definition: A theory T is inconsistent iff it is not the case that T is consistent. Definition: A theory T is complete iff for every formula P either T proves P or T proves ~P. Definition: A theory T is incomplete iff it is not the case that T is complete. Definition: A formula P is independent in a theory T iff there is no proof in T of P and there is no proof in T of ~P. Theorem: A theory T is inconsistent iff every formula P is provable in T and every formula P is disprovable in T. Theorem: A theory T is consistent iff there is a formula P that is not provable in T and there is a formula Q that is not disprovable in T. Theorem: A theory T is complete iff every formula P is such that either P is provable in T or P is disprovable in T. Theorem: A formula P is independent in a theory T iff P is not provable in T and P is not disprovable in T. Theorem: If there is a formula P that is independent in a theory T, then T is consistent. NO CONTRADICTIONS THERE! Now just see if you can get your mouth to say all that. Just try to form the words with your mouth and hear yourself say them. I mean, if you won't allow yourself to UNDERSTAND them, then at least you can prove to yourself that you know how to SAY them. MoeBlee
From: MoeBlee on 30 Jul 2010 17:43 On Jul 30, 4:28 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > 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. > > LOL. And you're so funny MoeBlee! Didn't I say you're a clown. > I actually did! Well, SOME comic relief is needed to break the tedium that is a conversation with you. MoeBlee
From: Nam Nguyen on 30 Jul 2010 17:43
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? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt ----------------------------------------------------------- |