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/?member_id=8660244&id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
