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: 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
