In a Consistent System, Can a True Sentence Imply a False One?
in [Logic]
|
Prev: when math defines the boundary between finite versus infinite at 10^500 #696 Correcting Math
Next: FLT like 4Color Mapping, Poincare C. and Kepler Packing #697 Correcting Math
From: Nam Nguyen on 11 Aug 2010 20:55 Nam Nguyen wrote: > 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. Let me add some more details on the nature of the relativity of the theory rPA. First let me repeat the group B axioms: > Group B: For each pair of distinct sub-languages, e.g. and without loss > of generality, L1 = (0, S', +', *', <') and L2 = (0, S'', +'', > *'', <''), there be this axiom: > > S'S'S'S'...S'S'(0) = S''S''S''S''S''...S''S''S''(0) By generalizing, extending, or otherwise altering Group B axioms, we'd in effect how relative rPA would be. For instance, if we have only this formula in the group: Axy[(x=y) -> (S'(x)=S''(y))] then rPA would exhibit zero relativity, so to speak. Otoh, the previous above formula involving equality of 2 terms of different lengths is too broad and would cause rPA to exhibit "chaotic" relativity. So, unless otherwise indicated, the terms involved in group B axioms above are those that are odd prime numbers w.r.t. their perspective multiplications (*' and *'' in this case of L1 and L2). In addition, we also add the following axiom to group B: (S'0 = S''0) /\ (S'S'0 = S''S''0), again for L1 and L2. Now then let's first let the notations: L(0,S,+,*,<), L(0,S_1,+_1,*_1,<_1), ..., L(0,S_n,+_n,*_n,<_n),... be variables for the sub-languages of L(rPA) of mentioned above. Then let's also have a defined unary symbol prime(x) axiomatized as [prime(x) could also be a new symbol in L(rPA) if it's so desired]: prime_n(x) -> prime(x) where prime_n(x) would express x being a prime number in the sub-language denoted by the variable, say, L(0,+,*,<). In addition let's have the following definitions in the meta level, with L(0,+,*,<) as a sub-language variable. Def1: A formula F written in L(0,+,*,<) is an absolute theorem of rPA (or absolutely unprovable in rPA) iff F', obtained by replacing all symbols of L(0,+,*,<) by _any_ particular sub-language value, is a canonical theorem of rPA. For example, 0=0, Ax[~(x=0) -> (x > 0)] Def2: A formula F written in L(0,+,*,<) is said to be absolutely unprovable iff its negation is absolutely provable. Def3: A formula F written in L(0,+,*,<) is said to be a relative theorem of rPA, iff there's a value for L(0,+,*,<) in which its corresponding F' is provable in rPA, and another value in which its corresponding F' is unprovable in rPA. Now then we'd come across some of the relative theorems of the theory rPA: Exy[(prime(x) /\ prime(y) /\ (x=y)) -> (x < y)] Exy[(prime(x) /\ prime(y) /\ (x=y)) -> (x*x = y*y)] [That's to say if no technical errors have occurred here, of course.] -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 11 Aug 2010 21:03 Nam Nguyen wrote: > Nam Nguyen wrote: >> 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. > > Let me add some more details on the nature of the relativity of the > theory rPA. First let me repeat the group B axioms: > > >> Group B: For each pair of distinct sub-languages, e.g. and without loss >> of generality, L1 = (0, S', +', *', <') and L2 = (0, S'', +'', >> *'', <''), there be this axiom: >> >> S'S'S'S'...S'S'(0) = S''S''S''S''S''...S''S''S''(0) > > By generalizing, extending, or otherwise altering Group B axioms, we'd > in effect how relative rPA would be. For instance, if we have only > this formula in the group: > > Axy[(x=y) -> (S'(x)=S''(y))] > > then rPA would exhibit zero relativity, so to speak. Otoh, the previous > above formula involving equality of 2 terms of different lengths is too > broad and would cause rPA to exhibit "chaotic" relativity. So, unless > otherwise indicated, the terms involved in group B axioms above are those > that are odd prime numbers w.r.t. their perspective multiplications > (*' and *'' in this case of L1 and L2). In addition, we also add the > following axiom to group B: (S'0 = S''0) /\ (S'S'0 = S''S''0), again > for L1 and L2. > > Now then let's first let the notations: > > L(0,S,+,*,<), L(0,S_1,+_1,*_1,<_1), ..., L(0,S_n,+_n,*_n,<_n),... > > be variables for the sub-languages of L(rPA) of mentioned above. Then > let's also have a defined unary symbol prime(x) axiomatized as [prime(x) > could also be a new symbol in L(rPA) if it's so desired]: > > prime_n(x) -> prime(x) > > where prime_n(x) would express x being a prime number in the sub-language > denoted by the variable, say, L(0,+,*,<). > > In addition let's have the following definitions in the meta level, > with L(0,+,*,<) as a sub-language variable. > > Def1: A formula F written in L(0,+,*,<) is an absolute theorem > of rPA (or absolutely unprovable in rPA) iff F', obtained > by replacing all symbols of L(0,+,*,<) by _any_ particular > sub-language value, is a canonical theorem of rPA. > > For example, 0=0, Ax[~(x=0) -> (x > 0)] > > Def2: A formula F written in L(0,+,*,<) is said to be absolutely > unprovable iff its negation is absolutely provable. > > Def3: A formula F written in L(0,+,*,<) is said to be a relative > theorem of rPA, iff there's a value for L(0,+,*,<) in which > its corresponding F' is provable in rPA, and another value > in which its corresponding F' is unprovable in rPA. > > Now then we'd come across some of the relative theorems of the > theory rPA: > > Exy[(prime(x) /\ prime(y) /\ (x=y)) -> (x < y)] > Exy[(prime(x) /\ prime(y) /\ (x=y)) -> (x*x = y*y)] Please cancel these 2 formulas. That's not what I wanted to say. More to come. Thanks. > > [That's to say if no technical errors have occurred here, of course.] > -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 11 Aug 2010 23:23 <Note> Let me do a repost of some of what was said before, for typo correction and for better clarity. </Note> **** Let me add some more details on the nature of the relativity of the theory rPA. First let me restate or re-edit both group A and B axioms: Group A: The familiar PA axioms for each sub-language that's isomorphic to L(PA). Group B: For each pair of distinct sub-languages, e.g. and without loss of generality, L1 = (0, S', +', *', <') and L2 = (0, S'', +'', *'', <''), there be these axioms: (1) (S'0 = S''0) /\ (S'S'0 = S''S''0) This formula expresses the uniqueness of all the successors of 0 (one successor per each sub-language). This uniqueness is represented by 1, which is a defined symbol of L(rPA). Similarly, this formula also represents the uniqueness of all the successors of 1 (one successor per each sub-language). This uniqueness is represented by 2, which is another defined symbol of L(rPA). Note that consequently, being a 1 or 2 is being expressed so in all said sub-languages of L(rPA), which also means we can't tell which sub-language is the underlying sub-language when referring to 1 or 2. (2) S'S'S'S'...S'S'(0) = S''S''S''S''S''...S''S''S''(0) where the numeral on the left of = is of different length (in term of FOL language symbols) than the one on the right, and where all the numerals are odd prime numbers w.r.t. their perspective multiplications (*' and *'' in this case of L1 and L2). Note also that, by altering Group B axiom (2) to the following: Axy[(x=y) -> (S'(x)=S''(y))] we would in effect reduce rPA to "no relativity", as well as making (1) redundant. On the other hand if we're not careful we could make the relativity of rPA very "chaotic" and undesired. So (1) and (2) seem to be a right balance of relativity "flavors". Now then let's define some more notations and let: L(0,S,+,*,<), L(0,S_1,+_1,*_1,<_1), ..., L(0,S_n,+_n,*_n,<_n),... be variables for the sub-languages of L(rPA) of mentioned above. Then let's also have a defined unary symbol prime(x) axiomatized as: prime_n(x) -> prime(x) where prime_n(x) would express x being a prime number in the sub-language denoted by the variable, say, L(0,+,*,<). As in the case of symbols 1 and 2, however, being prime(x) is being expressed so in all said sub-languages of L(rPA), which also means we can't tell which sub-language is the underlying sub-language when referencing to prime(x). This is in fact at the heart of expressibility of relativity of rPA. In addition let's also have the following definitions in the meta level, with L(0,+,*,<) as a sub-language variable. Def1: A formula F written in L(0,+,*,<) is an absolute theorem of rPA (or absolutely provable in rPA) iff F', obtained by replacing all non-logical symbols of F in L(0,+,*,<) by by counterpart symbols in _any_ particular sub-language value, is a canonical theorem of rPA. For example, 0=0, Ax[~(x=0) -> (x > 0)] Def2: A formula F written in L(0,+,*,<) is said to be absolutely unprovable iff its negation is absolutely provable. Def3: A formula F written in L(0,+,*,<) is said to be a relative theorem of rPA, iff there's a value for L(0,+,*,<) in which its corresponding F' is provable in rPA, and another value in which its corresponding F' is unprovable in rPA. Now then we'd come across one example of the relative theorems of the theory rPA: Exy[(prime(x) /\ prime(y) /\ (x=y)) -> (Az[z=2+x -> z=2+y])]
From: Nam Nguyen on 12 Aug 2010 15:39 Nam Nguyen wrote: > Let me add some more details on the nature of the relativity of the > theory rPA. Let me continue with more relevant comments on rPA. Lorentz Transformation: ======================= Fwiw, Group B axiom-schema (2) could be called Lorentz Transformation because they look reminiscent to the SR counterpart. In fact, on meta level, for a given 2 numbers x and x' that are prime(x) and prime(x'), their mutual distance is relative to which sub-language is the underlying one, because x or x' themselves each also has a relative distance to the common number 0. Relationship to ZF and to The Uncertainty Principle: ==================================================== In principle, the concept of the natural numbers could be taken as a priori where these numbers are just numbers and NOT sets. But naturally we could use ZF _sets_ for these "numbers". Now then suppose for a moment that it's impossible to know the arithmetic truth value of cGC (and at least intuitively it looks virtually certain that the impossibility is real), what could we say the concept of "set" viz-aviz ZF's encoding the natural numbers? There's a certain physics law about conservation of mass-energy which stipulates that if there's a mass it could be converted to energy but the original amount of mass/energy should remains constant. So would the impossibility of knowing the truth value of cGC (if such impossibility exists) just go away just because the naturals are now ZF _sets_ ? Imho, the answer is a "No": it will get converted to an unpleasant properties of ZF, namely the _uncertainty_ of the epsilon relation of L(ZF). Indeed, if we let c-GC be the set of counterexamples of GC, then there are _3_ distinct possibilities for the size of the set c-GC: zero length (i.e. empty set), finite but non-zero, and aleph_0. Iow, if it's impossible to know the arithmetic truth value of cGC, then it's uncertain that the set c-GC be empty, finite-non-empty, or infinite! The physics Uncertainty Principle says something to the effect that given certain properties such as a particle's position and momentum, "the more precisely one property is measured, the less precisely the other can be measure". It seems to me there's a reminiscence in FOL reasoning with the concept of the naturals as a mind-pacifying icon in the foundation: - The more we're certain that GC is true the less certain we'd know which consistent extension of PA would become inconsistent when being added with cGC as a new axiom. - The more we're certain that GC is false, the less certain we'd know whether or not c-GC is of the size alpeh_0. Arguably, Relativity and Uncertainty are foundational to the physical world, and arguably the concept of the Natural Numbers and that of the ZF set-hood are foundational in FOL reasoning. Is it just a coincidence that the 2 foundational concepts in FOL share some similar traces with the 2 foundational properties of the real physical world? Somehow I have a feeling this isn't a coincidence! Though you might of course have a different opinion. Regards, Nam Nguyen -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 13 Aug 2010 01:51
Nam Nguyen wrote: > Nam Nguyen wrote: > >> Let me add some more details on the nature of the relativity of the >> theory rPA. > > Let me continue with more relevant comments on rPA. > > Lorentz Transformation: > ======================= > > Fwiw, Group B axiom-schema (2) could be called Lorentz Transformation > because they look reminiscent to the SR counterpart. In fact, on meta > level, for a given 2 numbers x and x' that are prime(x) and prime(x'), > their mutual distance is relative to which sub-language is the underlying > one, because x or x' themselves each also has a relative distance to > the common number 0. > > Relationship to ZF and to The Uncertainty Principle: > ==================================================== > > In principle, the concept of the natural numbers could be taken as > a priori where these numbers are just numbers and NOT sets. But > naturally we could use ZF _sets_ for these "numbers". Now then > suppose for a moment that it's impossible to know the arithmetic > truth value of cGC (and at least intuitively it looks virtually > certain that the impossibility is real), what could we say the > concept of "set" viz-aviz ZF's encoding the natural numbers? > > There's a certain physics law about conservation of mass-energy > which stipulates that if there's a mass it could be converted to > energy but the original amount of mass/energy should remains constant. > So would the impossibility of knowing the truth value of cGC (if > such impossibility exists) just go away just because the naturals > are now ZF _sets_ ? Imho, the answer is a "No": it will get converted > to an unpleasant properties of ZF, namely the _uncertainty_ of the > epsilon relation of L(ZF). Indeed, if we let c-GC be the set of > counterexamples of GC, then there are _3_ distinct possibilities > for the size of the set c-GC: zero length (i.e. empty set), finite > but non-zero, and aleph_0. Iow, if it's impossible to know the > arithmetic truth value of cGC, then it's uncertain that the set > c-GC be empty, finite-non-empty, or infinite! > > The physics Uncertainty Principle says something to the effect that > given certain properties such as a particle's position and momentum, > "the more precisely one property is measured, the less precisely the > other can be measure". It seems to me there's a reminiscence in > FOL reasoning with the concept of the naturals as a mind-pacifying > icon in the foundation: > > - The more we're certain that GC is true the less certain we'd know > which consistent extension of PA would become inconsistent when > being added with cGC as a new axiom. That's actually not true: (PA + GC) would be such system. The statement should have been: > - The more we're certain that GC is true the less certain we'd know > which consistent extension of PA, having ~GC as a theorem, would > become inconsistent when being added with cGC as a new axiom. > > - The more we're certain that GC is false, the less certain we'd > know whether or not c-GC is of the size alpeh_0. > > Arguably, Relativity and Uncertainty are foundational to the physical > world, and arguably the concept of the Natural Numbers and that > of the ZF set-hood are foundational in FOL reasoning. > > Is it just a coincidence that the 2 foundational concepts in FOL > share some similar traces with the 2 foundational properties of > the real physical world? Somehow I have a feeling this isn't a > coincidence! Though you might of course have a different opinion. > > Regards, > > Nam Nguyen > -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt ----------------------------------------------------------- |