Charles, defining "formal system" is just as difficult as defining "number"-- in fact, those problems are basically equivalent. Godel made use of that to make his proof work (or at least that's one way of looking at it). So, if you claim that the concept "number" is dependent on what formal system it is defined in, shouldn't you also say the same thing of "formal system"?
So, for example, we might agree to interpret "number" as "Peano-arithmatic number" for the purpose of some discussion. But, we might still disagree on how to interpret Peano Arithmetic. I might say, "statement X isn't derivable from PA, by transfinite induction", and you might reply, "Hey, no, you're not allowed to use transfinite induction." Then we would need to settle which logic to interpret PA in: maybe you convince me that we've got to stick to robinson arithmetic. But then, I use some line of reasoning about the properties of Robinson arithmetic, and we've got to settle on an even higher system to resolve the argument... The point is, we've got to make some actual choice of default at some point. By arguing with anything I want to at every level, I'm using the classical-type default, which is essentially to use the strongest system available. Perhaps you would choose one of these logics that Godel's theorem fails for as your default. If we can figure out what default is normatively ideal, then in my opinion we've made important headway. I think I found the logics you're referring to? Looks *very* interesting. http://en.wikipedia.org/wiki/Self-verifying_theories --Abram On Fri, Oct 31, 2008 at 2:26 AM, Charles Hixson <[EMAIL PROTECTED]> wrote: > It all depends on what definition of number you are using. If it's > constructive, then it must be a finite set of numbers. If it's based on > full Number Theory, then it's either incomplete or inconsistent. If it's > based on any of several subsets of Number Theory that don't allow > incompleteness to be proven (or even described) then the numbers are > precisely this which is included in that subset of the theory. > > Number Theory is the one with the largest (i.e., and infinite number) of > unprovable theories about numbers of the variations that I have been > considering. My point in the just prior post is that numbers are precisely > that item which the theory you are using to describe them says they are, > since they are artifacts created for computational convenience, as opposed > to direct sensory experiences of the universe. > > As such, it doesn't make sense to say that a subset of number theory leaves > more facts about numbers undefined. In the subsets those aren't facts about > numbers. > > Abram Demski wrote: >> >> Charles, >> >> OK, but if you argue in that manner, then your original point is a >> little strange, doesn't it? Why worry about Godelian incompleteness if >> you think incompleteness is just fine? >> >> "Therefore, I would assert that it isn't that it leaves "*even more* >> about numbers left undefined", but that those characteristics aren't >> in such a case properties of numbers. Merely of the simplifications >> an abstractions made to ease computation." >> >> In this language, what I'm saying is that it is important to examine >> the "simplifications and abstractions", and discover how they work, so >> that we can "ease computation" in our implementations. >> >> --Abram >> >> On Thu, Oct 30, 2008 at 7:58 PM, Charles Hixson >> <[EMAIL PROTECTED]> wrote: >> >>> >>> If you were talking about something actual, then you would have a valid >>> point. Numbers, though, only exist in so far as they exist in the theory >>> that you are using to define them. E.g., if I were to claim that no >>> number >>> larger than the power-set of energy states within the universe were >>> valid, >>> it would not be disprovable. That would immediately mean that only >>> finite >>> numbers were valid. >>> >>> P.S.: Just because you have a rule that could generate a particular >>> number >>> given a larger than possible number of steps doesn't mean that it is a >>> valid >>> number, as you can't actually ever generate it. I suspect that infinity >>> is >>> primarily a computational convenience. But one shouldn't mistake the >>> fact >>> that it's very convenient for meaning that it's true. Or, given Occam's >>> Razor, should one? But Occam's Razor only detects provisional truths, >>> not >>> actual ones. >>> >>> If you're going to be constructive, then you must restrict yourself to >>> finitely many steps, each composed of finitely complex reasoning. And >>> this >>> means that you must give up both infinite numbers and irrational numbers. >>> To do otherwise means assuming that you can make infinitely precise >>> measurements (which would, at any rate, allow irrational numbers back >>> in). >>> >>> Therefore, I would assert that it isn't that it leaves "*even more* about >>> numbers left undefined", but that those characteristics aren't in such a >>> case properties of numbers. Merely of the simplifications an >>> abstractions >>> made to ease computation. >>> > > > > ------------------------------------------- > agi > Archives: https://www.listbox.com/member/archive/303/=now > RSS Feed: https://www.listbox.com/member/archive/rss/303/ > Modify Your Subscription: > https://www.listbox.com/member/?&; > Powered by Listbox: http://www.listbox.com > ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
