Hmmm. I think that some of our miscommunication might have been due to the
fact that you seem to be talking about two things while I think that I'm
talking about third . . . .
I believe that *meaning* is constructed.
I believe that truth is absolute (within a given context) and is a proper
subset of meaning.
I believe that proof is constructed and is a proper subset of truth (and
therefore a proper subset of meaning as well).
So, fundamentally, I *am* a constructivist as far as meaning is concerned
and take Gödel's theorem to say that meaning is not completely defined or
definable.
Since I'm being a constructionist about meaning, it would seem that your
statement that
A constructivist would be justified in asserting the equivalence of
Gödel's incompleteness theorem and Tarski's undefinability theorem,
would mean that I was "correct" (or, at least, not wrong) in using Gödel's
theorem but probably not as clear as I could have been if I'd used Tarski
since an additional condition/assumption (constructivism) was required.
So, interchanging the two theorems is fully justifiable in some
intellectual circles! Just don't do it when non-constructivists are
around :).
I guess the question is . . . . How many people *aren't* constructivists
when it comes to meaning? Actually, I get the impression that this mailing
list is seriously split . . . .
Where do you fall on the constructivism of meaning?
----- Original Message -----
From: "Abram Demski" <[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Sunday, October 26, 2008 10:00 PM
Subject: Re: [agi] constructivist issues
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
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