How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Charlie-Boo on 29 Jun 2010 12:18 On Jun 29, 11:56 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 29, 8:06 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > On Jun 29, 8:28 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > George Greene <gree...(a)email.unc.edu> writes: > > > > I really don't think that the model existence theorem is going to leap > > > > out at him here. > > > > Pretty much any elementary text contains an account of the set theoretic > > > construction of the various number systems. > > E.g., Enderton's 'Elements Of Set Theory'. > > > > It's a trivial exercise to > > > verify the axioms of PA hold in the structure of naturals. > > The only slightly non-trivial part is the induction schema, but it > falls right into place if one simply starts to do it. > > > >Combined with > > > the soundness of first-order logic this immediately yields the > > > consistency of PA. > > Soundness, e.g. demonstrated in Enderton's 'A Mathematical > Introduction To Logic'. Enderton only makes a comment that ZFC can prove the consistency of PA (pg. 270) with no details given. Is this why people are up in arms claiming that ZFC can prove PA consistent, even with no references to it - because it is conventional wisdom among academia? Then you really do have a mess - a claim that nobody has ever demonstrated. > > Can that be carried out in ZFC? > > YES! > > > Who has? > > I have! Many people have. > > But am I going to type ALL the formulas that go into this? No, I'm > not, since I'm not being paid for that kind of thing. How about telling me the title of a book or article in which PA is proved consistent using only ZFC? > If you wish to think I'm merely claiming that I've done such proofs, > then fine, you may believe as you wish. But, on the other hand, if you > wish to do just a basic investigation to see how you could do such a > proof yourself, then you only need to read a basic textbook in set > theory and one in mathematical logic. I've read numerous but none have a proof of PA consistency using only ZFC. > Just look at Enderton's two books. Not there. > If you wish to consider that evasion, then so be it. But a reasonable > person would understand that when such basic material is in the basic > textbooks of the subject, Then why can no one (including yourself) name one? > then posters may be reasonable in not > peforming the labor of typing out of textbooks when the person > inquiring could just get a book himself and read it. How can I do that when the only references are bogus? C-B > MoeBlee
From: MoeBlee on 29 Jun 2010 12:22 On Jun 29, 11:01 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > On Jun 29, 11:48 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 29, 7:58 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > Charlie-Boo <shymath...(a)gmail.com> writes: > > > > How about when you said that Gentzen proved PA consistent using ZFC? > > > > No one ever proves anything using ZFC in the sense of producing formal > > > derivations. Rather, they prove mathematical theorems using mathematical > > > principles formalized in ZFC. By certain elementary considerations we > > > know the formalizations of such theorems are formally derivable in ZFC. > > > One quibble: There are people who do produce (even if only exercises) > > displays of certain formal ZFC derivations, as well as those who > > generate such derivations with computer programs, etc. > > That's right - or so goes the party line. It's not a quibble - it > contradicts his "No one" claim. > > > Other than that, I think what you wrote is a well stated needed > > summary that should be part of a standing FAQ: > > > Ordinarily, > > What do you mean by "ordinarily"? When would it be ordinary and when > is it no? If this is well stated, then that should be well defined. It's not itself a mathematical assertion. It's an assertion about how mathematicians write and speak. By 'ordinarily', I would include the journal articles, books, textbooks, and lectures given by mathematicians working in classical mathematics. I've never seen one of those that consisted merely of displays of purely formal derivations in ZFC. Rather, the proofs given consist of combinations of mathematical symbols and English such that said proofs can be formalized in ZFC. If you read a good book on first order predicate calculus then a good book on set theory, you will understand. I can't impart all of the knowledge of a couple of books in a few posts. MoeBlee
From: MoeBlee on 29 Jun 2010 12:30 On Jun 29, 11:18 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > On Jun 29, 11:56 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > On Jun 29, 8:06 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > > On Jun 29, 8:28 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > > George Greene <gree...(a)email.unc.edu> writes: > > > > > I really don't think that the model existence theorem is going to leap > > > > > out at him here. > > > > > Pretty much any elementary text contains an account of the set theoretic > > > > construction of the various number systems. > > > E.g., Enderton's 'Elements Of Set Theory'. > > > > > It's a trivial exercise to > > > > verify the axioms of PA hold in the structure of naturals. > > > The only slightly non-trivial part is the induction schema, but it > > falls right into place if one simply starts to do it. > > > > >Combined with > > > > the soundness of first-order logic this immediately yields the > > > > consistency of PA. > > > Soundness, e.g. demonstrated in Enderton's 'A Mathematical > > Introduction To Logic'. > > Enderton only makes a comment that ZFC can prove the consistency of PA > (pg. 270) with no details given. Enderton proves the soundness theorem. That was my point in that particular mention. > Is this why people are up in arms claiming that ZFC can prove PA > consistent, even with no references to it - because it is conventional > wisdom among academia? Then you really do have a mess - a claim that > nobody has ever demonstrated. I need to stop posting with you. You're not listening. It's not just convential wisdom. Rather, it is a simple exercise anyone who knows about the subject can carry out. I have carried it out myself. I'm not going to type out my notes for you, because you could do the exercise yourself. > > > Can that be carried out in ZFC? > > > YES! > > > > Who has? > > > I have! Many people have. > > > But am I going to type ALL the formulas that go into this? No, I'm > > not, since I'm not being paid for that kind of thing. > > How about telling me the title of a book or article in which PA is > proved consistent using only ZFC? Just take the needed information from Enderton's two books. Then perform the proof as a simple exercise. > > If you wish to think I'm merely claiming that I've done such proofs, > > then fine, you may believe as you wish. But, on the other hand, if you > > wish to do just a basic investigation to see how you could do such a > > proof yourself, then you only need to read a basic textbook in set > > theory and one in mathematical logic. > > I've read numerous but none have a proof of PA consistency using only > ZFC. > > > Just look at Enderton's two books. > > Not there. Just take the needed information and do it as an exercise. I have. YOU as much as stated the overall outline when you noted how N is shown in ZFC and then the axioms of PA proven in ZFC (or whatever exactly you said). > > If you wish to consider that evasion, then so be it. But a reasonable > > person would understand that when such basic material is in the basic > > textbooks of the subject, > > Then why can no one (including yourself) name one? All you need to complete the exercise is provided you in Enderton's two books. > > then posters may be reasonable in not > > peforming the labor of typing out of textbooks when the person > > inquiring could just get a book himself and read it. > > How can I do that when the only references are bogus? I really need to stop with you. MoeBlee
From: Charlie-Boo on 29 Jun 2010 12:30 On Jun 29, 12:13 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 29, 10:25 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > > > > > On Jun 29, 10:55 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > On Jun 28, 9:07 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > > > On Jun 28, 12:24 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > And it's easy enough to see that if a theory has a model then that > > > > > theory is consistent. > > > > > Is that an axiomatic proof in ZFC? > > > > Plain Z-regularity proves that if a theory has a model then the theory > > > is consistent. It's quite simple; you would come up with it yourself > > > on just a moment's reflection. > > > Sorry, but I don't know what proof you have in mind, so I can't > > determine how the Axiom of Regularity would play a role. > > It DOESN'T play a role, which is why I took it out. I asked what axiom is essential and would be needed to carry out the proof in PA. I thought you were answering that. So what is the answer? > In other words, we > can prove in Z set theory even without the axiom of regularity > (possibly without certain other axioms? but I've never done such > detailed bookkeeping, as my only claim is that Z-R is SUFFICIENT). > > As to the proof. Would you just TRY to do it in your mind one time? I did. You can't just say the Axiom of Infinity provides a model as you have to also prove that implies consistency. You claim to know how, so be a mathematician and substantiate your claim. > If > you still can't see it, then, if I'm feeling generous, I'll outline it > for you. As to showing an exact sequence of primitive formulas of the > language of Z, no, that's just a chore. I don't know what you're referring to. I did ask for the statement of the theorem in ZFC, but nobody has come up with that either. So in summary, 1. ZFC can prove PA consistent - it's easy and lots of people have done it. 2. Nobody can give a reference to its being done. 3. Nobody can describe the proof that has been done in ZFC. 4. Nobody can give even the ZFC expression for the theorem itself. In other words, business as usual. C-B > We don't do that. Instead we > study enough first order logic so that we know how to convert informal > mathematical arguments into a primitive language and to see (by taking > progressively larger and larger chunks of argument) how one would > formalize into primitive language if one had to do such a thing; but > actually writing the pure formal primitive formulas is not required > for purposes of mathematical communication among adequately informed > people. > > If you wish to take that as evasion, then so be it. Meanwhile, though, > I have trained myself to see how to formalize, as have many other > people with whom I may talk about these matters; I'm not going to > perform unpaid labor of typing formula after formula for you, when you > could just as easily see that it can be done if you only studied a > book on first order logic and one on set theory. > > > What is the proof and how is Regularity essential? > > You're mixed up. Regularity is NOT needed. That's why I put Z- > regularity, which means "Z without the axiom of regularity". > > MoeBlee- Hide quoted text - > > - Show quoted text -
From: Charlie-Boo on 29 Jun 2010 12:42
On Jun 29, 12:20 pm, James Burns <burns...(a)osu.edu> wrote: > Charlie-Boo wrote: > > On Jun 29, 10:34 am, Chris Menzel > > <cmen...(a)remove-this.tamu.edu> wrote: > >>On Tue, 29 Jun 2010 03:34:12 -0700 (PDT), Charlie-Boo > >><shymath...(a)gmail.com> said: > >>>On Jun 29, 12:18 am, Chris Menzel > >>><cmen...(a)remove-this.tamu.edu> wrote: > >>>>The best known approach uses a mapping that Ackermann > >>>>defined from the hereditarily finite sets into N > > >>>There are too many sets to map them 1-to-1 with > >>>the natural numbers. > > >>Apparently you have yet to master the semantic role > >>of adjectives. > > > Ok, then tell me. What is the semantic role of adjectives? > > Part of the semantic role of adjectives is to cause > "hereditarily finite sets" to mean something different > from "sets". I was referring to sets in general, that ZFC can define more than aleph-1 sets - but I'm not so sure of that, actually. > Now that I've got you started, why don't > you go ahead and see if you can figure the rest of it out. > >http://blog.mrm.org/wp-content/uploads/2007/09/wizardofoz.jpg > > Who are you supposed to be? The Wizard? Dorothy? Toto? Anyone who makes claims without substantiation is the wizard - the big fakes that we all know about. C-B > It can't be Toto, because Toto would have at least tried > to figure out what the semantic role of adjectives was. > Jim Burns- Hide quoted text - > > - Show quoted text - |