>> Any formal system that contains some basic arithmetic apparatus equivalent >> to http://en.wikipedia.org/wiki/Peano_axioms is doomed to be incomplete with >> respect to statements about numbers... that is what Godel originally >> showed...
Oh. Ick! My bad phrasing. WITH RESPECT TO NUMBERS should have been WITH RESPECT TO THE DEFINITION OF NUMBERS since I was responding to "Numbers are not well-defined and can never be". Further, I should not have said "information about numbers" when I meant "definition of numbers". <two radically different things> Argh! = = = = = = = = So Ben, how would you answer Abram's question "So my question is, do you interpret this as meaning "Numbers are not well-defined and can never be" (constructivist), or do you interpret this as "It is impossible to pack all true information about numbers into an axiom system" (classical)?" Does the statement that a formal system is "incomplete with respect to statements about numbers" mean that "Numbers are not well-defined and can never be". = = = = = = = (Semi-)Retraction - maybe? (mostly for Abram). Ick again! I was assuming that we were talking about constructivism as in Constructivist epistemology (http://en.wikipedia.org/wiki/Constructivist_epistemology). I have just had Constructivism (mathematics) pointed out to me (http://en.wikipedia.org/wiki/Constructivism_(mathematics)) All I can say is "Ick!" I emphatically do not believe "When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence". = = = = = = = = I'm quitting and going home now to avoid digging myself a deeper hole :-) Mark PS. Ben, I read and, at first glance, liked and agreed with your argument as to why uncomputable entities are useless for science. I'm going to need to go back over it a few more times though. :-) ----- Original Message ----- From: Ben Goertzel To: [email protected] Sent: Tuesday, October 28, 2008 5:55 PM Subject: Re: [agi] constructivist issues Any formal system that contains some basic arithmetic apparatus equivalent to http://en.wikipedia.org/wiki/Peano_axioms is doomed to be incomplete with respect to statements about numbers... that is what Godel originally showed... On Tue, Oct 28, 2008 at 2:50 PM, Mark Waser <[EMAIL PROTECTED]> wrote: That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete Yes, any formal system is doomed to be incomplete. Emphatically, NO! It is not true that "any formal system" is doomed to be incomplete WITH RESPECT TO NUMBERS. It is entirely possible (nay, almost certain) that there is a larger system where the information about numbers is complete but that the other things that the system describes are incomplete. So my question is, do you interpret this as meaning "Numbers are not well-defined and can never be" (constructivist), or do you interpret this as "It is impossible to pack all true information about numbers into an axiom system" (classical)? Hmmm. From a larger reference framework, the former claimed-to-be-constructivist view isn't true/correct because it clearly *is* possible that numbers may be well-defined within a larger system (i.e. the "can never be" is incorrect). Does that mean that I'm a classicist or that you are mis-interpreting constructivism (because you're attributing a provably false statement to constructivists)? I'm leaning towards the latter currently. ;-) ----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]> To: <[email protected]> Sent: Tuesday, October 28, 2008 5:02 PM Subject: Re: [agi] constructivist issues Mark, That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete, meaning there will be statements that can be constructed purely by reference to numbers (no red cats!) that the system will fail to prove either true or false. So my question is, do you interpret this as meaning "Numbers are not well-defined and can never be" (constructivist), or do you interpret this as "It is impossible to pack all true information about numbers into an axiom system" (classical)? Hmm.... By the way, I might not be using the term "constructivist" in a way that all constructivists would agree with. I think "intuitionist" (a specific type of constructivist) would be a better term for the view I'm referring to. --Abram Demski On Tue, Oct 28, 2008 at 4:13 PM, Mark Waser <[EMAIL PROTECTED]> wrote: Numbers can be fully defined in the classical sense, but not in the constructivist sense. So, when you say "fully defined question", do you mean a question for which all answers are stipulated by logical necessity (classical), or logical deduction (constructivist)? How (or why) are numbers not fully defined in a constructionist sense? (I was about to ask you whether or not you had answered your own question until that caught my eye on the second or third read-through). ------------------------------------------- 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/?&; Powered by Listbox: http://www.listbox.com -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] "A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects." -- Robert Heinlein ------------------------------------------------------------------------------ agi | Archives | Modify Your Subscription ------------------------------------------- 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
