Torkel Franzen on truth
in [Logic]
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From: george on 7 Dec 2007 17:06 On Dec 6, 5:54 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > Hold on, hold on. You've rather ripped this out of context from Dan's > lecture notes. He two paragraphs earlier introduces the assumption > that we are dealing with systems of arithmetic S which are *sound* > (and hence consistent). Come ON. That is a lame dodge on YOUR part. HE is the one who changed the context by drawing THE DIRECT ANALOGY WITH NON-Euclidean geometry! NObody deprecates non-Euclidean geometries as "unsound"! That parlance in the general community IS INDEFENSIBLE. The relevant discovery is that "unsound" models of PA EVEN *can* Exist at all! Since the only reason GIT is important is that it proves THAT, for ANYbody to be trying to confine the discussion to "sound" systems IS IDIOTIC. The whole import of GIT is that PA+~Con(PA) IS CONSISTENT. For anybody to be whining "but it's unsound" is just that -- whining. The IMPORTANT point here is NOT, as Isaacson has alleged, a DIFFERENCE between the way the parallel postulate relates to models of geometry and the way G(PA) relates to models of PA. The IMPORTANT part is the SIMILARITY between the way they relate -- since neither of them is provable from the (other) axioms, BOTH of them are true in some models and false in others. > And on *that* assumption, which I take to be > still in force when we comments on the proof he sketches, In the section I quoted, he iS NOT sketching any proof -- I am quoting from AFTER that. In the section I quoted, he is drawing a (wrong) analogy between Con(PA) and the parallel postulate. > the > canonical Gödel sentence will indeed be true on the standard > interpretation. But proving that is quite beyond the scope of the material. You have to go somewhere like ZF and epsilon_0-induction to prove THAT.
From: george on 7 Dec 2007 17:13 > > On Dec 5, 5:27 pm, kleptomaniac6...(a)hotmail.com wrote: > > > In the case of FLT, if every positive integer n had the > > > property that it was not the z in a counterexample to FLT, so to > > > speak. I replied: > > No, NOT *so* to speak. That is incorrect. > > The correct way of speaking it, assuming you have an exponentiation > > operator, is > > An[n>2-->Axyz[~x^n+y^n=z^n] ]. IT IS *n* that must have or lack the > > property. You have to check the n. On Dec 6, 10:38 pm, p...(a)see.signature.invalid (Pierre Asselin) wrote: > But you can eliminate the inner quantifiers if you use tuples: > > every integer has the property that { > if you decode a quadruple <x,y,z,n> from it, > then > } But that is missing the point,too. You could just say Axyzn[n<3 or x^n + y^n != z^n]. Whether that is or isn't Pi-1 is, precisely as you say, not even worth arguing about because you can encode the tuple. The fact that that is possible is a good reason for writing it the way I write it (with 1 quantifier symbol instead of 4). It also bears stressing that this is not even a tuple, that the order DOES NOT matter, that any of the other 23 orders of the 4 variables would be THE SAME sentence. What DOES matter and what IS the point is that a "cuonterexample to FLT" is going to be THE *n* and NOT the "z", as klept0 was mis- alleging. And it also matters that you have to have some sort of Pi-1 definition of exponentiation to get this into the language of PA. I presume the original Godel proof uses the chinese remainder theorem among other things to manage that.
From: george on 7 Dec 2007 17:16 On Dec 7, 1:56 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > It does seem, indeed, that some of > the suspect claims people make e.g. about our supposedly not be able > know that PA is consistent have nothing much to do the philosophy of > mathematics itself, Society advances 1 funeral at a time. Doubts about the consistency of systems satisfying the hypotheses of G1 come from 1 very well-known place: infinity. That is especially relevant since proofs HAVE to be finite. G1 is part of a cluster of results implying, basically, that a pre- requisite for the existence of a finitary description of an infinite thing is that the infinite thing be recursive. If it is merely (like 1st-order theories) rec.enum. instead, then there is definitionally always doubt. In BOTH directions.
From: george on 7 Dec 2007 17:22 On Dec 7, 2:07 pm, pbopa...(a)gmail.com wrote: > Well, it seems to me that this last assertion is completely wrong. "2 > + 2 = 4" seems more certain to me than the consistency of PA, and I > imagine the same is true for you, as well. Well, 2+2=4 is a theorem of PA. Con(PA) is not. So of course it seems more ceratin. > After all, the consistency > of PA - that a particular logico-mathematical system does not ever > produce among its deductions "not 0 = 0" - presumably depends on an > argument for you to believe it. "2 + 2 = 4" does not. Yes, actually, it does. That theorem has a proof just like any other. The fact that you went through the argument in 1st grade does not end the existence of the argument. > There is plenty of reason to believe that water boils at 100 degrees > centigrade, and no reason to believe otherwise. That hardly means > that it is "as certain as anything." There is plenty of reason not to bring the physical world into this.
From: george on 7 Dec 2007 17:24
On Dec 6, 5:54 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > Oh? Why does it force us to ask that?? Actually I do know Dan, and of > course we'd both agree with Torkel Franzen's point. Oh, stop lying. The whole reason I quoted it is that Isaacson is directly contradicting TF's point here. |