Mark, After some thought...
A constructivist would be justified in asserting the equivalence of Godel's incompleteness theorem and Tarski's undefinability theorem, based on the idea that truth is constructable truth. Where classical logicians take Godels theorem to prove that provability cannot equal truth, constructivists can take it to show that provability is not completely defined or definable (and neither is truth, since they are the same). So, interchanging the two theorems is fully justifiable in some intellectual circles! Just don't do it when non-constructivists are around :). --Abram On Sat, Oct 25, 2008 at 6:18 PM, Mark Waser <[EMAIL PROTECTED]> wrote: > OK. A good explanation and I stand corrected and more educated. Thank you. > > ----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]> > To: <[email protected]> > Sent: Saturday, October 25, 2008 6:06 PM > Subject: Re: [agi] constructivist issues > > >> Mark, >> >> Yes. >> >> I wouldn't normally be so picky, but Godel's theorem *really* gets >> misused. >> >> Using Godel's theorem to say made it sound (to me) as if you have a >> very fundamental confusion. You were using a theorem about the >> incompleteness of proof to talk about the incompleteness of truth, so >> it sounded like you thought "logically true" and "logically provable" >> were equivalent, which is of course the *opposite* of what Godel >> proved. >> >> Intuitively, Godel's theorem says "If a logic can talk about number >> theory, it can't have a complete system of proof." Tarski's says, "If >> a logic can talk about number theory, it can't talk about its own >> notion of truth." Both theorems rely on the Diagonal Lemma, which >> states "If a logic can talk about number theory, it can talk about its >> own proof method." So, Tarski's theorem immediately implies Godel's >> theorem: if a logic can talk about its own notion of proof, but not >> its own notion of truth, then the two can't be equivalent! >> >> So, since Godel's theorem follows so closely from Tarski's (even >> though Tarski's came later), it is better to invoke Tarski's by >> default if you aren't sure which one applies. >> >> --Abram >> >> On Sat, Oct 25, 2008 at 4:22 PM, Mark Waser <[EMAIL PROTECTED]> wrote: >>> >>> So you're saying that if I switch to using Tarski's theory (which I >>> believe >>> is fundamentally just a very slightly different aspect of the same >>> critical >>> concept -- but unfortunately much less well-known and therefore less >>> powerful as an explanation) that you'll agree with me? >>> >>> That seems akin to picayune arguments over phrasing when trying to simply >>> reach general broad agreement . . . . (or am I misinterpreting?) >>> >>> ----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]> >>> To: <[email protected]> >>> Sent: Friday, October 24, 2008 5:29 PM >>> Subject: Re: [agi] constructivist issues >>> >>> >> >> >> ------------------------------------------- >> 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 > ------------------------------------------- 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
