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: <agi@v2.listbox.com>
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).
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agi
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