How Can ZFC/PA do much of Math - it Can't Even Prove PA isConsistent(EASYPROOF)
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From: William Hale on 3 Jul 2010 11:01 In article <4C2F47AC.57FFDAAD(a)gmail.com>, herbzet <herbzet(a)gmail.com> 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? I thought the question was whether "PA is consistent" is doubtful or not? I could have an inconsistent theory and not doubt the proof on one of its stated theorems. In fact, incorrect proofs can be stated in an inconsistent theory: the theorem would of course be true, but that does not make any proof valid. In other words, in an inconsistent theory, I could have no doubt about a proof being correct, or I could have no doubt about a proof being wrong.
From: Marshall on 3 Jul 2010 11:05 On Jul 3, 7:22 am, herbzet <herb...(a)gmail.com> wrote: > > What is PA + (1)? The successor to PA. Marshall
From: herbzet on 3 Jul 2010 12:45 William Hale 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? > > I thought the question was whether "PA is consistent" is doubtful or not? I thought so too. > I could have an inconsistent theory and not doubt the proof on one of > its stated theorems. In fact, incorrect proofs can be stated in an > inconsistent theory: the theorem would of course be true, but that does > not make any proof valid. In other words, in an inconsistent theory, I > could have no doubt about a proof being correct, or I could have no > doubt about a proof being wrong. Not following you. Do you find the proof of the infinitude of primes unconvincing as to whether there is an infinitude of primes? Are you, perhaps, uncertain of the truth of the premises? -- hz
From: herbzet on 3 Jul 2010 12:46 Marshall wrote: > herbzet wrote: > > > > What is PA + (1)? > > The successor to PA. The more I look at this, the funnier it gets. Make it stop!
From: MoeBlee on 3 Jul 2010 16:49
On Jul 3, 11:36 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > Alan Smaill wrote: > >> Nam Nguyen <namducngu...(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_ ? > > I don't believe MoeBlee has asserted that -- > he can answer for himself. Not only did I not assert it, I even before reminded Nam that my remarks in that regard were to convey Aatu's view not necessarily mine, and even as that was clear in those remarks. It's hopeless with Nam. MoeBlee |