X-Message-Number: 23927 Date: Sun, 18 Apr 2004 23:17:31 -0700 From: Mike Perry <> Subject: Re: Goedelian problems Robert Ettinger, #23916: >Mike Perry writes in part: > > >expanding the formal system to remove one [Goedelian] "problem" > >does not at all remove another, similar problem in the expanded system. > >The first "problem" is (say) of the type: "Sentence number one is >unprovable," while giving that sentence the label "number one." > >The new "problem" in the different system might be of the type: "Sentence >number two is unprovable," while giving that sentence the label "number two." > >Are these "different" problems, and does either have any significance beyond >quirks of language? The two problems are different in detail though basically similar. I think they have significance well beyond quirks of language, since, for instance, they demonstrate limitations in our ability to predict the behavior of computers. We'd like to know if a certain program is in an infinite loop. Maybe the computer is working on something important, whose outcome we must forfeit if we terminate execution too soon. Or maybe it will never find the answer so the sooner we stop the less we waste precious resources. (This assumes too that its thrashing, in absence of the successful outcome, will not be worth the cost, which will arguably hold sometimes.) This strikes me as a very practical concern. Goedel's result has a direct bearing on it, telling us we can't tell in general, in advance of "what happens, happens," and will have to trust our luck. One might have suspected this, yes, but having a formal argument is something more than speculation. This is not to deny there are other weaknesses in formal systems--which still can be quite powerful. Goedel's result also tells us that a mathematical theory of everything, along the lines mathematicians generally like to use (2-valued or n-valued logic, for instance), is impossible. No such theory can be comprehensive enough to do even the basic theory of integer arithmetic, and also complete and consistent. That's quite an accomplishment Mike Perry Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=23927