New essay on Goedel's Incompleteness Theorem
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From: Nam Nguyen on 26 Sep 2009 17:59 Marshall wrote: > On Sep 26, 10:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Since Godel's theorem assumes the existences of the naturals as a model of, >> say, Q, Godel's work basically assumes the consistency of Q. > > That doesn't sound right to me. Does the proof even mention > Robinson arithmetic? Doesn't Q date from the 1950s and > aren't Godel's proofs from the 1930s? I took the liberty here to equate Q to PM, which Godel did mention. > > I suppose it is correct to say that the technique of encoding > formulas depends on the existence of arithmetic. Does > that seem like a weakness? Yes. I posted this at least a couple of times in the past. But basically, for Godel's theorem to work with *all* formal systems (meeting certain requirements of course), a reservoir of _infinite_ number of primes would be needed, _without_ a constraints as to what value they might be. Which means that if half of the world assumes GC be true and uses an associated infinite set S1 of primes and the other half uses infinite set associated with cGC (the counter GC) then one half would be incorrect for sure. But the whole world wouldn't which half is correct and which half is incorrect. The weakness of using arithmetic without the *rigor* backup of syntactical formalism vis-a-vis axioms is we'd be reasoning on our own _intuition_, _outside the reasoning-protection_ of rule of inference! > Do you have doubts about arithmetic on the naturals? Yes for sure. And I think any human being should. > Not some axiomatization of them, I mean; > the actual naturals and their operators as a model. In mathematics (via FOL), a model is always a model - a reflection - of a formal system. If you don't really know what the _reflect-ee_ really be, you wouldn't know some of the assertions you might make!
From: Nam Nguyen on 26 Sep 2009 18:13 Nam Nguyen wrote: > Marshall wrote: >> On Sep 26, 10:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> Since Godel's theorem assumes the existences of the naturals as a >>> model of, >>> say, Q, Godel's work basically assumes the consistency of Q. >> >> That doesn't sound right to me. Does the proof even mention >> Robinson arithmetic? Doesn't Q date from the 1950s and >> aren't Godel's proofs from the 1930s? > > I took the liberty here to equate Q to PM, which Godel did mention. >> >> I suppose it is correct to say that the technique of encoding >> formulas depends on the existence of arithmetic. Does >> that seem like a weakness? > > Yes. I posted this at least a couple of times in the past. But basically, > for Godel's theorem to work with *all* formal systems (meeting certain > requirements of course), a reservoir of _infinite_ number of primes would > be needed, _without_ a constraints as to what value they might be. Caveat: I certainly didn't mean Godel didn't know how to manipulate prime numbers (or arithmetic in general). The weakness in his work is _not the tactical_ technicality of arithmetic recursion, encoding, etc... It's his _strategic_ assumption that knowledge of a syntactical consistency could be equated to the intuitive knowledge of a model. For the finite cases, perhaps. But no human could do the equating all the times, being finite beings. (Remember "finite" is a foundation of rules of inferences, proofs, formulas, etc...) > > Which means that if half of the world assumes GC be true and uses an > associated infinite set S1 of primes and the other half uses infinite > set associated with cGC (the counter GC) then one half would be incorrect > for sure. But the whole world wouldn't which half is correct and which > half is incorrect. > > The weakness of using arithmetic without the *rigor* backup of syntactical > formalism vis-a-vis axioms is we'd be reasoning on our own _intuition_, > _outside the reasoning-protection_ of rule of inference! > >> Do you have doubts about arithmetic on the naturals? > > Yes for sure. And I think any human being should. > >> Not some axiomatization of them, I mean; >> the actual naturals and their operators as a model. > > In mathematics (via FOL), a model is always a model - a reflection - of > a formal system. If you don't really know what the _reflect-ee_ really be, > you wouldn't know some of the assertions you might make!
From: Marshall on 26 Sep 2009 20:06 On Sep 26, 2:59 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Sep 26, 10:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Since Godel's theorem assumes the existences of the naturals as a model of, > >> say, Q, Godel's work basically assumes the consistency of Q. > > > That doesn't sound right to me. Does the proof even mention > > Robinson arithmetic? Doesn't Q date from the 1950s and > > aren't Godel's proofs from the 1930s? > > I took the liberty here to equate Q to PM, which Godel did mention. > > > > > I suppose it is correct to say that the technique of encoding > > formulas depends on the existence of arithmetic. Does > > that seem like a weakness? > > Yes. I posted this at least a couple of times in the past. But basically, > for Godel's theorem to work with *all* formal systems (meeting certain > requirements of course), a reservoir of _infinite_ number of primes would > be needed, _without_ a constraints as to what value they might be. There is certainly an infinite number of primes. I don't know what "without constraints as to what they might be" means. The set of primes that I'm familiar with has constraints on it. For example, it has the constraint that 4 is not a member. This doesn't present a problem for using primes in encoding statements, however. (And anyway, aren't there other possible encoding techniques besides those using primes? I could be wrong.) > Which means that if half of the world assumes GC be true and uses an > associated infinite set S1 of primes and the other half uses infinite > set associated with cGC (the counter GC) then one half would be incorrect > for sure. But the whole world wouldn't which half is correct and which > half is incorrect. The set of primes does not depend on whether GC is true or not. There does not exist various different sets of all prime natural numbers; there is just the one. Exactly one of GC and !GC is true in the natural numbers. We don't know which one it is, but that doesn't matter when one is doing arithmetic. If I want to know what 2+2 is, do you say 4, or do you say "it depends"? > The weakness of using arithmetic without the *rigor* backup of syntactical > formalism vis-a-vis axioms is we'd be reasoning on our own _intuition_, > _outside the reasoning-protection_ of rule of inference! > > > Do you have doubts about arithmetic on the naturals? > > Yes for sure. And I think any human being should. Heh. I think any human being should not. > > Not some axiomatization of them, I mean; > > the actual naturals and their operators as a model. > > In mathematics (via FOL), a model is always a model - a reflection - of > a formal system. If you don't really know what the _reflect-ee_ really be, > you wouldn't know some of the assertions you might make! Again, my view is exactly the opposite. Models are essential and foundational. We can do a lot of mathematics with just models and without formal theories. (In fact, formal systems are a relatively recent mathematical development.) In contrast, a theory that has no model is not something that has much if any utility. Marshall
From: Nam Nguyen on 27 Sep 2009 02:57 Marshall wrote: > On Sep 26, 2:59 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Sep 26, 10:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Since Godel's theorem assumes the existences of the naturals as a model of, >>>> say, Q, Godel's work basically assumes the consistency of Q. >>> That doesn't sound right to me. Does the proof even mention >>> Robinson arithmetic? Doesn't Q date from the 1950s and >>> aren't Godel's proofs from the 1930s? >> I took the liberty here to equate Q to PM, which Godel did mention. >> >> >> >>> I suppose it is correct to say that the technique of encoding >>> formulas depends on the existence of arithmetic. Does >>> that seem like a weakness? >> Yes. I posted this at least a couple of times in the past. But basically, >> for Godel's theorem to work with *all* formal systems (meeting certain >> requirements of course), a reservoir of _infinite_ number of primes would >> be needed, _without_ a constraints as to what value they might be. > > There is certainly an infinite number of primes. I don't know what > "without constraints as to what they might be" means. The set > of primes that I'm familiar with has constraints on it. For example, > it has the constraint that 4 is not a member. This doesn't present > a problem for using primes in encoding statements, however. All that simply meant the set of primes must be infinite to enable the encoding for a theory with an arbitrary number of non-logical symbols, but the set has no constraints on what particular prime numbers must be in it (again, as long as the set is infinite). > > (And anyway, aren't there other possible encoding techniques > besides those using primes? I could be wrong.) Don't know for sure but my guess would be "No". >> Which means that if half of the world assumes GC be true and uses an >> associated infinite set S1 of primes and the other half uses infinite >> set associated with cGC (the counter GC) then one half would be incorrect >> for sure. But the whole world wouldn't which half is correct and which >> half is incorrect. > > The set of primes does not depend on whether GC is true or > not. There does not exist various different sets of all prime > natural numbers; there is just the one. All what was said here is there would be 2 different infinite sub-sets of primes, but nobody could tell which one be the good set to use. > > Exactly one of GC and !GC is true in the natural numbers. I meant cGC and not !GC, where cGC df= "There are infinite counter examples of GC". > We don't know which one it is, but that doesn't matter when > one is doing arithmetic. The truth or falsehood of GC or cGC is supposed to be an arithmetic one. So in this case, it does matter to know if GC, cGC, !GC is true or false. > If I want to know what 2+2 is, do > you say 4, or do you say "it depends"? The truth of 2+2=4 require finite knowledge. What would happen if you want to know the _arithmetic_ truth/falsehood of GC, cGC, or !GC? >>> Not some axiomatization of them, I mean; >>> the actual naturals and their operators as a model. >> In mathematics (via FOL), a model is always a model - a reflection - of >> a formal system. If you don't really know what the _reflect-ee_ really be, >> you wouldn't know some of the assertions you might make! > > Again, my view is exactly the opposite. Models are essential > and foundational. That view is incorrect, given the finite nature of rules of inferences and proofs, which are foundations of FOL reasoning. > We can do a lot of mathematics with just > models and without formal theories. Given that we're clueless for so long about the arithmetic truth value of one single formula like GC? > (In fact, formal systems are a relatively recent mathematical development.) That's why before reasoning was plagued by paradoxes! > In contrast, > a theory that has no model is not something that has much if > any utility. But who has said anything about praising inconsistent theories (that would have no models)?
From: Marshall on 27 Sep 2009 09:32
On Sep 26, 11:57 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Sep 26, 2:59 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Marshall wrote: > >>> On Sep 26, 10:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> Since Godel's theorem assumes the existences of the naturals as a model of, > >>>> say, Q, Godel's work basically assumes the consistency of Q. > >>> That doesn't sound right to me. Does the proof even mention > >>> Robinson arithmetic? Doesn't Q date from the 1950s and > >>> aren't Godel's proofs from the 1930s? > >> I took the liberty here to equate Q to PM, which Godel did mention. > > >>> I suppose it is correct to say that the technique of encoding > >>> formulas depends on the existence of arithmetic. Does > >>> that seem like a weakness? > >> Yes. I posted this at least a couple of times in the past. But basically, > >> for Godel's theorem to work with *all* formal systems (meeting certain > >> requirements of course), a reservoir of _infinite_ number of primes would > >> be needed, _without_ a constraints as to what value they might be. > > > There is certainly an infinite number of primes. I don't know what > > "without constraints as to what they might be" means. The set > > of primes that I'm familiar with has constraints on it. For example, > > it has the constraint that 4 is not a member. This doesn't present > > a problem for using primes in encoding statements, however. > > All that simply meant the set of primes must be infinite to enable the > encoding for a theory with an arbitrary number of non-logical symbols, > but the set has no constraints on what particular prime numbers must > be in it (again, as long as the set is infinite). Well, that describes the situation we have, so I see no difficulty here. > > (And anyway, aren't there other possible encoding techniques > > besides those using primes? I could be wrong.) > > Don't know for sure but my guess would be "No". > > >> Which means that if half of the world assumes GC be true and uses an > >> associated infinite set S1 of primes and the other half uses infinite > >> set associated with cGC (the counter GC) then one half would be incorrect > >> for sure. But the whole world wouldn't which half is correct and which > >> half is incorrect. > > > The set of primes does not depend on whether GC is true or > > not. There does not exist various different sets of all prime > > natural numbers; there is just the one. > > All what was said here is there would be 2 different infinite sub-sets of > primes, but nobody could tell which one be the good set to use. So you are talking about proper subsets. But encoding uses the whole set, so again your point does not affect the situation with regards to using primes for encoding. > > Exactly one of GC and !GC is true in the natural numbers. > > I meant cGC and not !GC, where cGC df= "There are infinite counter examples > of GC". > > > We don't know which one it is, but that doesn't matter when > > one is doing arithmetic. > > The truth or falsehood of GC or cGC is supposed to be an arithmetic one. > So in this case, it does matter to know if GC, cGC, !GC is true or false. > > > If I want to know what 2+2 is, do > > you say 4, or do you say "it depends"? > > The truth of 2+2=4 require finite knowledge. What would happen if you > want to know the _arithmetic_ truth/falsehood of GC, cGC, or !GC? These are all cute little rhetorical techniques you are using, but the fact remains that we can add, subtract, multiply and divide natural numbers with no trouble, without knowing the truth of, or even having heard of, Goldbach's Conjecture. We could do so just as well had Goldbach never been born. Just because we don't have infinite knowledge doesn't mean we can't add. > >>> Not some axiomatization of them, I mean; > >>> the actual naturals and their operators as a model. > >> In mathematics (via FOL), a model is always a model - a reflection - of > >> a formal system. If you don't really know what the _reflect-ee_ really be, > >> you wouldn't know some of the assertions you might make! > > > Again, my view is exactly the opposite. Models are essential > > and foundational. > > That view is incorrect, given the finite nature of rules of inferences > and proofs, which are foundations of FOL reasoning. Which is more important is something of a philosophical point. And I think theories are pretty cool and all. But they are unnecessary for many things, whereas you can't do squat without a model. > > We can do a lot of mathematics with just > > models and without formal theories. > > Given that we're clueless for so long about the arithmetic truth value > of one single formula like GC? Of course. Obviously. > > (In fact, formal systems are a relatively recent mathematical development.) > > That's why before reasoning was plagued by paradoxes! > > > In contrast, > > a theory that has no model is not something that has much if > > any utility. > > But who has said anything about praising inconsistent theories (that > would have no models)? This just shows you are overemphasizing the importance of theories. In summary, all you have done is point out that we don't have infinite knowledge. So what? That doesn't identify any problem, difficulty, or flaw with arithmetic or with Godel's proofs. Marshall |