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 12:53 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 > > -- > hz
From: Nam Nguyen on 3 Jul 2010 13:06 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? > >> 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. At least not until they know, say, the DNA of their "mother". FWIW, that's actually the story of one of the infamous Emperors of China, i.e., Qin Shi Huang (thought about his "father" instead of "mother")!
From: herbzet on 3 Jul 2010 13:15
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. -- hz |