How Can ZFC/PA do much of Math - it Can't Even Prove PA isConsistent(EASYPROOF)
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From: Nam Nguyen on 3 Jul 2010 13:22 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)?
From: herbzet on 3 Jul 2010 13:42 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? Is there an end to these seemingly pointless questions? -- hz
From: Nam Nguyen on 3 Jul 2010 13:59 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! 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. 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, ...". Give it up. 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.
From: Nam Nguyen on 3 Jul 2010 14:02 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 do you ask such a simple question? > > 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?
From: Nam Nguyen on 3 Jul 2010 14:24
Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> herbzet wrote: >>> Nam Nguyen wrote: >>>> herbzet wrote: > ... >>>>> 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 do you ask such a simple question? > > You were asking about whether consistency was ' "seen" ', complete > with square quotes. That's not asking about proofs, its > asking about personal intuitions. > > And the response was *not* claiming that the formal system is > inconsistent (or consistent, for that matter). But that's my whole point! Why do people, such as MoeBlee, assert that PA is consistent "PERIOD." when they don't have a way to know that for a fact, and _"seeing" is not a fact_ ? Why couldn't they admit they _in fact_ don't know that PA is consistent? |