How Can ZFC/PA do much of Math - it Can't Even Prove PA isConsistent(EASYPROOF)
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From: herbzet on 3 Jul 2010 14:26 Nam Nguyen wrote: > herbzet wrote: > > Nam Nguyen wrote: > >> herbzet wrote: > >>> Nam Nguyen wrote: > >>>> herbzet wrote: > >>>>> Nam Nguyen wrote: > >>>>>> herbzet wrote: > >>>>>>> Nam Nguyen wrote: > >>>>>>>> herbzet wrote: > >>>>>>>>> The whole thing is nonsense, anyway. Clearly, PA > >>>>>>>>> is consistent, or at least, its consistency is > >>>>>>>>> at least as evident as the consistency of any > >>>>>>>>> system which purports to prove it. > >>>>>>>> Sometimes it's much ... much simpler and more logical, humble, > >>>>>>>> humanistic to admit we don't know what we can't know, rather > >>>>>>>> than pretending to possess some sort of an immortal knowledge. > >>>>>>>> > >>>>>>>> Suppose someone states "There are infinitely many universes > >>>>>>>> and each has harbored a planet with intelligent life in its history." > >>>>>>>> > >>>>>>>> If there actually are infinitely many universes we can't know such > >>>>>>>> fact. PA's consistency is like such a statement: if it's consistent, > >>>>>>>> you can't never know that. Period. > >>>>>>> I see a model for PA: the natural numbers. I conclude PA is consistent. > >>>>>>> > >>>>>>> I would say I know this to a mathematical certainty. > >>>>>> IOW, mathematical reasoning is just a gambling that never ends! > >>>>> You find something doubtful in the proof of the infinitude of primes? > >>>> There are a lot of things I never find doubtful, such as the proof > >>>> Ax[~(Sx=0)] in PA. But there are also a lot of theories in which not > >>>> only Ax[~(Sx=0)] is provable but also Axy[x*y=0] is. By the natural > >>>> numbers, would you mean they collectively be a model of such theory? > >>> Do you mean, do I think the natural numbers are a model of the theory > >>> which has as axioms only the two formulas Ax[~Sx=0] and Axy[x*y=0]? > >> Right. > >> > >>>> Just _exactly_ what do you "see" as the natural numbers? > >>> 0, 1, 2, 3, ... > >> How would "0, 1, 2, 3, ..." _exactly_ prevent "Axy[x*y=0]" from being true? > > > > I would be honored to hear how from you. > > > > I await your reply with great anticipation! > > First of all, it's _you_ who believe about the _exactness_ of naturals > so it's your burden of proof to show "Axy[x*y=0]" is true! Funny, I don't think so. > Secondly, from your _mere_ "0, 1, 2, 3, ...", I could have: > > U = {0, 1, 2, 3, ...} > M = {..., <'*', {(x,y,0) | x,y are in U}>, ...} > > which means Axy[x*y=0] would true in M. First you ask me "How would "0, 1, 2, 3, ..." _exactly_ prevent "Axy[x*y=0]" from being true?" Then you show that "0, 1, 2, , ..." exactly do *not* prevent Axy[x*y=0] from being true. As I anticipated you would. You set a little trap, but I didn't fall in. I'm not mad, but I *am* a little bored. > Which means you didn't know what you were talking about when answering > my question "Just _exactly_ what do you "see" as the natural numbers?" > with the _mere_ "0, 1, 2, 3, ...". Funny, I don't think so. > Give it up. As you wish. > Not only you but nobody would know _exactly_ what the naturals > be. (Of course being circular doesn't count as being exact here!) > > > > >>>> If there's > >>>> a least one formula you couldn't [see?] its truth in the naturals, > >>>> in what sense do you "know" the naturals? > >>> I don't know my mother's blood type -- does that mean I don't know > >>> my own mother? > >> A lot of adopted children might not necessarily know the truth about > >> their being adopted as an infant. > > > > That is so. > > Good that you understand the analogy. > > > But my question was whether my lack of knowledge about my mother's > > blood type means that I don't know my own mother. > > So, is an adopting mother a "mother" in your definition? You got to be > _precise_ in arguing, you know. Why do I have to answer your questions, but you don't have to answer mine? That's not fair -- is it? -- hz
From: herbzet on 3 Jul 2010 14:50 Nam Nguyen wrote: > herbzet wrote: > > Nam Nguyen wrote: > >> herbzet wrote: > >>> Nam Nguyen wrote: > >>>> herbzet wrote: > >>>>> Nam Nguyen wrote: > >>>>>> herbzet wrote: > >>>>>>> Nam Nguyen wrote: > >>>>>>>> herbzet wrote: > >>>>>>>>> Nam Nguyen wrote: > >>>>>>>>>> herbzet wrote: > >>>>>>>>>>> The whole thing is nonsense, anyway. Clearly, PA > >>>>>>>>>>> is consistent, or at least, its consistency is > >>>>>>>>>>> at least as evident as the consistency of any > >>>>>>>>>>> system which purports to prove it. > >>>>>>>>>> Sometimes it's much ... much simpler and more logical, humble, > >>>>>>>>>> humanistic to admit we don't know what we can't know, rather > >>>>>>>>>> than pretending to possess some sort of an immortal knowledge. > >>>>>>>>>> > >>>>>>>>>> Suppose someone states "There are infinitely many universes > >>>>>>>>>> and each has harbored a planet with intelligent life in its history." > >>>>>>>>>> > >>>>>>>>>> If there actually are infinitely many universes we can't know such > >>>>>>>>>> fact. PA's consistency is like such a statement: if it's consistent, > >>>>>>>>>> you can't never know that. Period. > >>>>>>>>> I see a model for PA: the natural numbers. I conclude PA is consistent. > >>>>>>>>> > >>>>>>>>> I would say I know this to a mathematical certainty. > >>>>>>>> > >>>>>>>> While having such a "seeing", do you "see" if PA + (1) is consistent? > >>>>>>>> > >>>>>>>> I mean, how far can you go with such "seeing"? > >>>>>>> > >>>>>>> What is PA + (1)? > >>>>>> Let's define the 2 formulas in L(PA): > >>>>>> > >>>>>> pGC <-> "There are infinitely many examples of Goldbach Conjecture." > >>>>> I don't know what this means. If it means "There are an infinite > >>>>> number of even numbers that are the sum of two prime numbers" then > >>>>> it's provable in in PA [...] > >>>> That's what pGC means. Now back to my question: > >>>> > >>>> >>>> While having such a "seeing", do you "see" if PA + (1) is consistent? > >>>> > >>>> It's a simple technical question, of which the answer would be "Yes", or "No", > >>>> or "I don't know". > >>>> > >>>> Which one would be your answer? > >>>> > >>>>>> cGC <-> "There are infinitely many counter examples of Goldbach Conjecture." > >>>>>> > >>>>>> Then: > >>>>>> > >>>>>> (1) <-> pGC xor cGC > > >>> No -- I don't "see" if PA + (1) is consistent. > >> > >> Since inconsistency of a FOL formal system T is merely a finite > >> proof, you must have "seen" such a proof for PA + (1)? > > > > Must I have? > > Of course you must, if you want your technical statement to be credible. Why does my not seeing if PA + (1) is consistent imply that I have "seen" a proof of the inconsistency of PA + 1? > Why do you ask such a simple question? Because I was puzzled by your response, and I was hoping for some clarification from you. > > Is there an end to these seemingly pointless questions? > > Do you mean to say there's no end to questioning your credibility in > answering technical questions? No, I do not mean to say that. I mean to ask you to indicate the relevance of this series of seemingly pointless questions to my assertion that since I see a model of PA, I conclude, to what I consider a mathematical certainty, that PA is consistent. I've been patient, but I'm not infinitely patient. Do you have anything to say to this point? -- hz
From: Nam Nguyen on 3 Jul 2010 15:48 herbzet wrote: > > Nam Nguyen wrote: > >> Secondly, from your _mere_ "0, 1, 2, 3, ...", I could have: >> >> U = {0, 1, 2, 3, ...} >> M = {..., <'*', {(x,y,0) | x,y are in U}>, ...} >> >> which means Axy[x*y=0] would true in M. > > First you ask me "How would "0, 1, 2, 3, ..." _exactly_ prevent > "Axy[x*y=0]" from being true?" > > Then you show that "0, 1, 2, , ..." exactly do *not* prevent > Axy[x*y=0] from being true. > > As I anticipated you would. > > You set a little trap, but I didn't fall in. > > I'm not mad, but I *am* a little bored. Oh no, you were simply mistaken. The exact description of the naturals to you is a mere "0, 1, 2, 3, ..." which I pointed out that that IS NOT precise, since that's just a (universe) set at best. The naturals must be more than just a set of elements, surely you must know that. And I just gave you a demonstration that you were wrong about the naturals is just "0, 1, 2, 3, ...", because from that U, one could construct a model in which Axy[x*y=0] would be true. Hope that you now see my _simple_ counter argument to your "exact" "understanding/seeing" the naturals. >> So, is an adopting mother a "mother" in your definition? You got to be >> _precise_ in arguing, you know. > > Why do I have to answer your questions, but you don't have to answer mine? > > That's not fair -- is it? Fairness in technical Q/A requires your questions or premises have to be exact, precise. I'm sure you've heard of "Garbage in, Garbage out"!
From: herbzet on 3 Jul 2010 16:23 Nam Nguyen wrote: > herbzet wrote: > > Nam Nguyen wrote: > > > >> Secondly, from your _mere_ "0, 1, 2, 3, ...", I could have: > >> > >> U = {0, 1, 2, 3, ...} > >> M = {..., <'*', {(x,y,0) | x,y are in U}>, ...} > >> > >> which means Axy[x*y=0] would true in M. > > > > First you ask me "How would "0, 1, 2, 3, ..." _exactly_ prevent > > "Axy[x*y=0]" from being true?" > > > > Then you show that "0, 1, 2, , ..." exactly do *not* prevent > > Axy[x*y=0] from being true. > > > > As I anticipated you would. > > > > You set a little trap, but I didn't fall in. > > > > I'm not mad, but I *am* a little bored. > > Oh no, you were simply mistaken. Oh no, I'm definitely a little bored with your eristic. > The exact description of the naturals > to you is a mere "0, 1, 2, 3, ..." which I pointed out that that IS NOT > precise, since that's just a (universe) set at best. No, you didn't. > The naturals must > be more than just a set of elements, surely you must know that. Must they? > And I just gave you a demonstration that you were wrong about the > naturals is just "0, 1, 2, 3, ...", because from that U, one > could construct a model in which Axy[x*y=0] would be true. So the set of naturals can be the domain of models of more than one theory. So what? > Hope that you now see my _simple_ counter argument to your "exact" > "understanding/seeing" the naturals. > > >> So, is an adopting mother a "mother" in your definition? You got to be > >> _precise_ in arguing, you know. > > > > Why do I have to answer your questions, but you don't have to answer mine? > > > > That's not fair -- is it? > > Fairness in technical Q/A requires your questions or premises have to be > exact, precise. I'm sure you've heard of "Garbage in, Garbage out"! Well, yes, I have. Goodbye. -- hz
From: Nam Nguyen on 3 Jul 2010 16:48
herbzet wrote: > > Nam Nguyen wrote: >> herbzet wrote: >>> Nam Nguyen wrote: >>> >>>> Secondly, from your _mere_ "0, 1, 2, 3, ...", I could have: >>>> >>>> U = {0, 1, 2, 3, ...} >>>> M = {..., <'*', {(x,y,0) | x,y are in U}>, ...} >>>> >>>> which means Axy[x*y=0] would true in M. >>> First you ask me "How would "0, 1, 2, 3, ..." _exactly_ prevent >>> "Axy[x*y=0]" from being true?" >>> >>> Then you show that "0, 1, 2, , ..." exactly do *not* prevent >>> Axy[x*y=0] from being true. >>> >>> As I anticipated you would. >>> >>> You set a little trap, but I didn't fall in. >>> >>> I'm not mad, but I *am* a little bored. >> Oh no, you were simply mistaken. > > Oh no, I'm definitely a little bored with your eristic. > >> The exact description of the naturals >> to you is a mere "0, 1, 2, 3, ..." which I pointed out that that IS NOT >> precise, since that's just a (universe) set at best. > > No, you didn't. You seemed to have blasted cranks and trolls, but why your countering arguments have a trade-mark of theirs: short and no technical substance? > >> The naturals must >> be more than just a set of elements, surely you must know that. > > Must they? Of course they must according to _you_ below ... > >> And I just gave you a demonstration that you were wrong about the >> naturals is just "0, 1, 2, 3, ...", because from that U, one >> could construct a model in which Axy[x*y=0] would be true. > > So the set of naturals can be the domain of models of more than > one theory. How do you distinguish 2 different models of 2 different theories if all you have is just common U? > > So what? So you are wrong, in claiming you "see" the naturals, just by U = {1,2,3,...}, what ever that might be! You begin to argue like a crank! > >> Hope that you now see my _simple_ counter argument to your "exact" >> "understanding/seeing" the naturals. >> >>>> So, is an adopting mother a "mother" in your definition? You got to be >>>> _precise_ in arguing, you know. >>> Why do I have to answer your questions, but you don't have to answer mine? >>> >>> That's not fair -- is it? >> Fairness in technical Q/A requires your questions or premises have to be >> exact, precise. I'm sure you've heard of "Garbage in, Garbage out"! > > Well, yes, I have. It's always easy to claim something without proof! > > Goodbye. > Sure. I didn't ask for your arguing. |