In a Consistent System, Can a True Sentence Imply a False One?
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From: Aatu Koskensilta on 10 Aug 2010 17:21 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > In summary, that's what I meant to say when saying that Godel's work > doesn't apply to this kind of systems that would have this kind of > infinite non-logical but _sibling_ symbols. If that's not so, perhaps > an explanation could be offered? In order for the first incompleteness theorem (in its usual formulation) to apply to a theory T the theory must meet the following conditions: o The language of T is countable. o T is axiomatizable. o Robinson arithmetic is interpretable in T. Whether the language of T contains infinitely many symbols (and the nature of these symbols) is immaterial. Perhaps you can explain why you think these "sibling" symbols make any difference? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 10 Aug 2010 17:52 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> In summary, that's what I meant to say when saying that Godel's work >> doesn't apply to this kind of systems that would have this kind of >> infinite non-logical but _sibling_ symbols. If that's not so, perhaps >> an explanation could be offered? > > In order for the first incompleteness theorem (in its usual formulation) > to apply to a theory T the theory must meet the following conditions: > > o The language of T is countable. Sure. I believe L(rPA) (which is rL) can be considered as countable. > o Robinson arithmetic is interpretable in T. Sure. Given Robinson arithmetic is interpretable in each axiom set in group A, it looks to be it's also interpretable *in* rPA. > > Whether the language of T contains infinitely many symbols, and the > nature of these symbols, is immaterial. Perhaps you can explain why you > think these "sibling" symbols make any difference? Because of the axioms in group B: the other familiar arithmetic formal systems such as Q, or PA, does NOT have this kind of axioms arising out of infinitely many language symbols that are of the same n-ary type and are of the identical semantics! As I mentioned above, I don't have all the details yet but the encoding in Incompleteness is going to be trip-wired all over the places because of the axioms in this group. (Noter that group B axioms would allow us to to interpret an existentially identical prime number that would have different interpreted "arithmetic values" and yet, worse than that, we wouldn't be able to tell how different the values be!) -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 10 Aug 2010 18:35 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> In summary, that's what I meant to say when saying that Godel's work >>> doesn't apply to this kind of systems that would have this kind of >>> infinite non-logical but _sibling_ symbols. If that's not so, perhaps >>> an explanation could be offered? >> >> In order for the first incompleteness theorem (in its usual formulation) >> to apply to a theory T the theory must meet the following conditions: >> >> o The language of T is countable. > > Sure. I believe L(rPA) (which is rL) can be considered as countable. > >> o Robinson arithmetic is interpretable in T. > > Sure. Given Robinson arithmetic is interpretable in each axiom set > in group A, it looks to be it's also interpretable *in* rPA. >> >> Whether the language of T contains infinitely many symbols, and the >> nature of these symbols, is immaterial. Perhaps you can explain why you >> think these "sibling" symbols make any difference? > > Because of the axioms in group B: the other familiar arithmetic formal > systems such as Q, or PA, does NOT have this kind of axioms arising out > of infinitely many language symbols that are of the same n-ary type and > are of the identical semantics! > > As I mentioned above, I don't have all the details yet but the encoding > in Incompleteness is going to be trip-wired all over the places because > of the axioms in this group. (Noter that group B axioms would allow us to > to interpret an existentially identical prime number that would have > different interpreted "arithmetic values" and yet, worse than that, we > wouldn't be able to tell how different the values be!) Note that there's nothing to prevent us from considering L(PA) to be just as one of the sub-languages of L(rPA) that we mentioned above. So, in effect, there would be no more such a thing as " _the_ standard" language of arithmetic! And given that arithmetic truths have to be carried on the back of a language, group B axioms would mark the end of any notion of "standard-ness/absoluteness" in arithmetic truths, it would seem to me. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: MoeBlee on 11 Aug 2010 13:04 On Aug 10, 3:08 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > I'll concede my asserting that > Godel's work requires only finitely many non logical symbols for the > underlying T is too strong to be correct (and adequately portraying what > I'd like to say). Nicely done! Encore, encore! MoeBlee
From: Nam Nguyen on 11 Aug 2010 13:39
MoeBlee wrote: > On Aug 10, 3:08 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> I'll concede my asserting that >> Godel's work requires only finitely many non logical symbols for the >> underlying T is too strong to be correct (and adequately portraying what >> I'd like to say). > > Nicely done! Encore, encore! _As usual_ MoeBlee doesn't have much to say on the foundational issues and decides to act funny as a clown. What a surprise! -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt ----------------------------------------------------------- |