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 10:25 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." cGC <-> "There are infinitely many counter examples of Goldbach Conjecture." Then: (1) <-> pGC xor cGC
From: Nam Nguyen on 3 Jul 2010 10:35 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? Just _exactly_ what do you "see" as the natural numbers? If there's a least one formula you couldn't its truth in the naturals, in what sense do you "know" the naturals?
From: herbzet on 3 Jul 2010 12:42 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 -- and true. > cGC <-> "There are infinitely many counter examples of Goldbach Conjecture." > > Then: > > (1) <-> pGC xor cGC -- hz
From: herbzet on 3 Jul 2010 12:45 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]? > Just _exactly_ what do you "see" as the natural numbers? 0, 1, 2, 3, ... > 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? -- hz
From: Nam Nguyen on 3 Jul 2010 14:07
herbzet wrote: > > Nam Nguyen wrote: >> Nam Nguyen wrote: >>> herbzet wrote: >>>> Marshall wrote: >>>>> herbzet wrote: >>>>>> What is PA + (1)? >>>>> The successor to PA. >>>> The more I look at this, the funnier it gets. >>> Same here. >>> >>>> Make it stop! >>> Right. _Stop_ believing you "know" _exactly_ what the natural numbers be! >> Seriously, relativity in sciences (including mathematics) isn't an >> one-man conviction in "sci.logic", "sci.math". The mere mentioning >> of the 5th postulate, Hilbert-era's truth-equals-provability, SR, >> QM, should be a reminder that belief of any absoluteness in sciences >> is an ancient belief, which is no longer adequate for describing physical >> reality, or abstraction. >> >> If we scorn or laugh at the relativity of the standardness of a purported >> "model" of L(PA), a.k.a collectively as "the natural numbers", then we're >> no better that those who laughed at Riemann's ideas, at SR, at QM's uncertainty. >> At least those people had a valid excuse: they were in a different time in >> the past. We don't have such excuse! >> >> Seriously, all the nasty bickering aside, think about the whole thing logically. >> Think about the 4 reasoning Principles: >> >> - Principle of Consistency. >> - Principle of Compatibility. >> - Principle of Symmetry. >> - Principle of Humility. >> >> Would you think these are nonsensical principles honestly speaking? > > Probably. > > Who said anything about "absolute knowledge"? I think you're > tilting at a windmill. Oh, but by FOL definition of a formal system consistency, it must be either absolutely true or absolutely false that PA is consistent! (Ditto for inconsistency). Of course nobody should prevent you from saying: "I don't know"! See how easy when we're truthful to ourself! |