OK. A good explanation and I stand corrected and more educated. Thank you.
----- Original Message -----
From: "Abram Demski" <[EMAIL PROTECTED]>
To: <agi@v2.listbox.com>
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: <agi@v2.listbox.com>
Sent: Friday, October 24, 2008 5:29 PM
Subject: Re: [agi] constructivist issues
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