Excuse me, but I thought there were subsets of Number theory which were
strong enough to contain all the integers, and perhaps all the rational,
but which weren't strong enough to prove Gödel's incompleteness theorem
in. I seem to remember, though, that you can't get more than a finite
number of irrationals in such a theory. And I think that there are
limitations on what operators can be defined.
Still, depending on what you mean my Number, that would seem to mean
that Number was well-defined. Just not in Number Theory, but that's
because Number Theory itself wasn't well-defined.
Abram Demski wrote:
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
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