On Mar 31, 6:24 pm, barry barrett <barry3...@hotmail.com> wrote: > > On Mar 30, 1:04 am, barry barrett > > <barry3...@hotmail.com> wrote: > > > > As far as I've gotten there is no extension of the > > Godel Theorem like thet one you > > > describe. But I only got to the first proof of the > > Godel Thm (for Peano A w/out exponents) > > > Okay, but at this point I don't know what objection > > you still have to > > Godel's own generalization. > > We are interpreting Godel's generalization differently. I see it as a remark that, > potentially, points the way toward a more general proof (not as part of one). > > Godel's Thm is that any w-con system --- is incomplete.
No it's not. It's that any w-consistent, recursively axiomatized theory capable of a certain amount of arithmetic is incomplete.
Then, since we don't need to tie ourselves down to Godel himself, we usually state the Godel-Rosser result that generalizes from w- consistent to consistent.
> So we might use 1 & 2 as part of the argument towards this (but there are other ways > obviously), by showing that any w con sytem etc must obey conditions 1 & 2.
What was condition (2) again? Wasn't it satisfying Theorem V? In that case, I don't think that just any w-consistent system satisifes (2).
> But it seems that you are advocating a lesser and trivial theorem -that a system > satisfying 1 & 2 is incomplete- and mistaking it for the more general theorem which > assumes much less.
I'm not "advocating" weakening any theorem. It seems you're not following my remarks. Please look at what I said about a whole gradation of stronger, weaker, and incomparable conditions.
> >Returning to your other > > remark, yes, of > > course we can always look for certain properties that > > entail that > > theorem V holds. But that doesn't diminish that we > > can generalize, > > just as Godel mentions, to say that any consisten, > > recursively > > axiomatized system (not just Godel's system P > > specifically) of which > > theorem V holds is incomplete. > > But I would prefer to show the opposite. That, for any any w-con... Thm V holds. That, > it seems to me, is a proof of the general thm.
Godel did not make such a claim. I don't think it's even true. And it's not what was at stake when you began this matter. What was at stake then should be clear to have been resolved with us now: It is constructive and easy to execute the generalization that GODEL mentioned (and even for consistent as opposed to the stronger assumption of w-consistent).
> To recap, it seems to me that you are misinterpreting Godel's remark about conditions 1 > & 2. Or maybe I should say that, from my point of view, you are over-estimating the > power and purpose of them.
Nope. Please look at my exact remarks carefully and at Godel's remarks carefully. The only liberty I took was to switch from w-consistent to consistent. And that is not at all problematic.
> I don't believe that Godel felt that this remark (about conditions 1 & 2) was a step > away from a general proof;
And I didn't say that he did believe that. On the CONTRARY. (It seems to me that you've lost track in this matter. Please go over Godel's paper again, then my remarks).
> subject to obvious and trivial generalization. I read them as technical points about > one specific method of constructing an undecidable sentence in a given system -in a > sentence he is saying , "use the theory of recursve relations."
What he said is clear onto itself. Please reread it and its context.
