Torkel Franzen on truth
in [Logic]
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From: Daryl McCullough on 24 Nov 2007 10:12 Newberry says... >I am claiming that by proving that G is true we have proven G. Okay. But recall what G is. For any theory T capable of coding up sufficiently much of proof theory, you can form a sentence G such that G <-> not Pr_T(#G) where Pr_T(x) is the formalization in T of "x is the code of a statement provable in T". In other words, G is true if and only if G is not provable by theory T. It may be provable by *other* theories. >If we have proven this in a metalanguage then this system has its own G_1 >and its consistency is unprovable other than in even higher mata- >language. Therefore we have no consistency proof of PA at all. A proof is relative to a set of axioms. There are axioms that are strong enough to prove the consistency of PA, so your claim is false. If you are claiming that it doesn't count unless the theory can prove its own axioms to be consistent, well Godel showed that there is no system that can do that. >Then there is the manifest truth proof: >The axioms of PA are manifestly true >PA derivations preserve truth >Truth cannot be inconsistent with itself >i.e. PA is consistent >This proof is absolute and conclusive. It's an appeal to our intuitions. We don't prove the correctness of those intuitions, any more than ZFC proves the consistency of the axioms of ZFC. >It means that it is not formalizable. Well, we've gone into a complete circle. I say it's perfectly well formalizable. >For if it were it would either >not be conclusive It's *not* conclusive in the sense that you are wanting things to be conclusive. Nothing is. >just like in the meta-theory above or we would have >an inconsistency just as if we used P_universal(x,y). It means that >the human mind surpasses any machine. No, it doesn't. >How did we manage to surpass any machine? Since the proof is >conclusive we must have accomplished some non-formalizable analogy of >P_universal. Since it is not formalizable we are off the hook. We can >eat the turkey and have it. We have proven G without running into a >contradiction. > >You know what I am getting at? Yes, but it is completely wrong. There is no sense in which we can prove the Godel sentence for PA in any more conclusive fashion than a machine can. -- Daryl McCullough Ithaca, NY
From: Peter_Smith on 24 Nov 2007 10:23 On 24 Nov, 15:12, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: terrific good sense :-)))))))))))))))))
From: Newberry on 24 Nov 2007 12:43 On Nov 24, 7:12 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >I am claiming that by proving that G is true we have proven G. > > Okay. But recall what G is. For any theory T capable of coding > up sufficiently much of proof theory, you can form a sentence > G such that > > G <-> not Pr_T(#G) > > where Pr_T(x) is the formalization in T of "x is the code of > a statement provable in T". > > In other words, G is true if and only if G is not provable > by theory T. It may be provable by *other* theories. > > >If we have proven this in a metalanguage then this system has its own G_1 > >and its consistency is unprovable other than in even higher mata- > >language. Therefore we have no consistency proof of PA at all. > > A proof is relative to a set of axioms. "PA is consistent" means that in PA, for any P we will never derive P & ~P no matter how long and in what order we keep generating theorems. It is not relative to any axioms. It either is the case or not. This is what we are interested in proving. There are axioms that > are strong enough to prove the consistency of PA, so your claim > is false. If you are claiming that it doesn't count unless the > theory can prove its own axioms to be consistent, well Godel > showed that there is no system that can do that. So how can we possible arrive at the conclusion that PA is consistent? > > >Then there is the manifest truth proof: > >The axioms of PA are manifestly true > >PA derivations preserve truth > >Truth cannot be inconsistent with itself > >i.e. PA is consistent > >This proof is absolute and conclusive. > > It's an appeal to our intuitions. We don't > prove the correctness of those intuitions, > any more than ZFC proves the consistency of > the axioms of ZFC. > > >It means that it is not formalizable. > > Well, we've gone into a complete circle. I say > it's perfectly well formalizable. So which one is it? A) We don't prove the correctness of those intuitions B) it's perfectly well formalizable We can even try one hypothesis after another but not both at the same time. > >For if it were it would either > >not be conclusive > > It's *not* conclusive in the sense that you > are wanting things to be conclusive. Nothing is. > > >just like in the meta-theory above or we would have > >an inconsistency just as if we used P_universal(x,y). It means that > >the human mind surpasses any machine. > > No, it doesn't. > > >How did we manage to surpass any machine? Since the proof is > >conclusive we must have accomplished some non-formalizable analogy of > >P_universal. Since it is not formalizable we are off the hook. We can > >eat the turkey and have it. We have proven G without running into a > >contradiction. > > >You know what I am getting at? > > Yes, but it is completely wrong. There is no sense in which > we can prove the Godel sentence for PA in any more conclusive > fashion than a machine can. That's not what I am getting at. > -- > Daryl McCullough > Ithaca, NY
From: Daryl McCullough on 24 Nov 2007 14:25 Newberry says... > >On Nov 24, 7:12 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> A proof is relative to a set of axioms. > >"PA is consistent" means that in PA, for any P we will never derive P >& ~P no matter how long and in what order we keep generating theorems. >It is not relative to any axioms. It either is the case or not. This >is what we are interested in proving. But to "prove" something means to derive it from a set of axioms. Proof is relative to a set of axioms, but *truth* is not. So whether PA is consistent or not is a matter of fact, independent of any axioms. But whether we can *prove* that fact depends on what axioms we are using. >> There are axioms that >> are strong enough to prove the consistency of PA, so your claim >> is false. If you are claiming that it doesn't count unless the >> theory can prove its own axioms to be consistent, well Godel >> showed that there is no system that can do that. > >So how can we possible arrive at the conclusion that PA is consistent? Because it follows from other things we believe. There is no absolute sense in which we know it (or anything else). It's just that the consistency of PA follows from our best theories of mathematics. >So which one is it? >A) We don't prove the correctness of those intuitions >B) it's perfectly well formalizable Both are true. The consistency of PA follows from our intuitions. Our intuitions are pretty much formalizable. But we can't prove the correctness of those intuitions. >We can even try one hypothesis after another but not both at the same >time. > >> >For if it were it would either >> >not be conclusive >> >> It's *not* conclusive in the sense that you >> are wanting things to be conclusive. Nothing is. >> >> >just like in the meta-theory above or we would have >> >an inconsistency just as if we used P_universal(x,y). It means that >> >the human mind surpasses any machine. >> >> No, it doesn't. >> >> >How did we manage to surpass any machine? Since the proof is >> >conclusive we must have accomplished some non-formalizable analogy of >> >P_universal. Since it is not formalizable we are off the hook. We can >> >eat the turkey and have it. We have proven G without running into a >> >contradiction. >> >> >You know what I am getting at? >> >> Yes, but it is completely wrong. There is no sense in which >> we can prove the Godel sentence for PA in any more conclusive >> fashion than a machine can. > >That's not what I am getting at. Well, I don't know what you are getting at, then. But it is not true that Godel's theorem implies that humans surpass any machine. -- Daryl McCullough Ithaca, NY
From: Newberry on 24 Nov 2007 17:49
On Nov 24, 11:25 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Nov 24, 7:12 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> A proof is relative to a set of axioms. > > >"PA is consistent" means that in PA, for any P we will never derive P > >& ~P no matter how long and in what order we keep generating theorems. > >It is not relative to any axioms. It either is the case or not. This > >is what we are interested in proving. > > But to "prove" something means to derive it from a set of axioms. > Proof is relative to a set of axioms, but *truth* is not. So > whether PA is consistent or not is a matter of fact, independent > of any axioms. But whether we can *prove* that fact depends on > what axioms we are using. If the fact is independent of any axioms then I wonder what we need the axioms for. Let's forget about proofs then. How do we determine if "PA is consistent" is true? > > >> There are axioms that > >> are strong enough to prove the consistency of PA, so your claim > >> is false. If you are claiming that it doesn't count unless the > >> theory can prove its own axioms to be consistent, well Godel > >> showed that there is no system that can do that. > > >So how can we possible arrive at the conclusion that PA is consistent? > > Because it follows from other things we believe. There is no > absolute sense in which we know it (or anything else). It's > just that the consistency of PA follows from our best theories > of mathematics. Which theories? ZFC? We cannot prove that ZFC is consistent. At best we know it in the same sense we know PA is consistent. So I do not know what follows from what. > > >So which one is it? > >A) We don't prove the correctness of those intuitions > >B) it's perfectly well formalizable > > Both are true. The consistency of PA follows from ourive > intuitions. Our intuitions are pretty much formalizable. > But we can't prove the correctness of those intuitions. By formalization I mean a formal proof. We were discussing the fact that no matter how long we generate theorems of PA we will never derive a contradiction. How did we arrive at the knowledge that it is true? Is the method by which we arrived at it formalizable? (OK, there is a proof in ZFC that an arithmetical statement in some sense equivalent to "PA is consistent" is derivable from its axioms. That is not what we are interested in. We are interested in whether "PA is consistent" is true. How did we arive at the knowledge of this truth? Is the method we arrived at it formalizable?) I think you would agree with me that the answer is A. > > > > > >We can even try one hypothesis after another but not both at the same > >time. > > >> >For if it were it would either > >> >not be conclusive > > >> It's *not* conclusive in the sense that you > >> are wanting things to be conclusive. Nothing is. > > >> >just like in the meta-theory above or we would have > >> >an inconsistency just as if we used P_universal(x,y). It means that > >> >the human mind surpasses any machine. > > >> No, it doesn't. > > >> >How did we manage to surpass any machine? Since the proof is > >> >conclusive we must have accomplished some non-formalizable analogy of > >> >P_universal. Since it is not formalizable we are off the hook. We can > >> >eat the turkey and have it. We have proven G without running into a > >> >contradiction. > > >> >You know what I am getting at? > > >> Yes, but it is completely wrong. There is no sense in which > >> we can prove the Godel sentence for PA in any more conclusive > >> fashion than a machine can. > > >That's not what I am getting at. > > Well, I don't know what you are getting at, then. But it is > not true that Godel's theorem implies that humans surpass > any machine. > > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text - |