Vladimir Nesov, Then do you agree with my hypothetical extremist version of Mike?
(Aside: For the example we are talking about, it is totally necessary to stick the undecidable cases in F rather than T: if a Turing machine halts, then it is possible to prove that it halts (simply by running it for long enough). So if a Turing machine is one of those whose halting is formally undecidable, then it must not halt, because if it did then a proof of its halting would exist.) Hector Zenil, I do not think I understand you. Your argument seems similar to the following: "I do not see why Turing machines are necessary. If we can compute a function f(x) by some Turing machine, then we could compute it up to some value x=n. But we could construct a lookup table of all values f(0), f(1), f(2),... , f(n) which contains just as much information." Obviously the above is a silly argument, but I don't know how else to interpret you. A Turing machine can capture a finite number of the outputs of a hypercomputer. Does that in any way make the hypercomputer reducible to the Turing machine? Mike Archbold, It seems you've made a counterargument without meaning to. "When we make this transition, it seems to me that the shift is so radical that it is impossible to justify making the step, because as I mentioned it involves a surreptitious shift from quantity to quality." I maintain that the jump is justified. To me it is like observing the sequence "1, 2, 4, 8, 16, 32..." and concluding that each number is twice the previous. It is a jump from several quantities to a single quality. On Tue, Jun 17, 2008 at 4:35 PM, Vladimir Nesov <[EMAIL PROTECTED]> wrote: > On Tue, Jun 17, 2008 at 11:38 PM, Abram Demski <[EMAIL PROTECTED]> wrote: >> V. N., >> What is inhuman to me, is to claim that the halting problem is no >> problem on such a basis: that the statement "Turing machine X does not >> halt" only is true of Turing machines that are *provably* non-halting. >> And this is the view we are forced into if we abandon the reality of >> the uncomputable. >> > > Why, you can also mark up the remaining territory by "true" and > "false", these labels just won't mean anything there. Set up to sets, > T and F, place all true things in T, all false things in F, and all > unknown things however you like, but don't tell anybody how. Some > people like to place all unknown things in F, their call. > Mathematically it can be convenient, but really, even of "computable" > things you can't really compute that much, so the argument is void for > all practical concerns anyway. > > -- > Vladimir Nesov > [EMAIL PROTECTED] > > > ------------------------------------------- > agi > Archives: http://www.listbox.com/member/archive/303/=now > RSS Feed: http://www.listbox.com/member/archive/rss/303/ > Modify Your Subscription: http://www.listbox.com/member/?&; > Powered by Listbox: http://www.listbox.com > ------------------------------------------- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244&id_secret=106510220-47b225 Powered by Listbox: http://www.listbox.com
