It all depends on what definition of number you are using. If it's
constructive, then it must be a finite set of numbers. If it's based on
full Number Theory, then it's either incomplete or inconsistent. If
it's based on any of several subsets of Number Theory that don't allow
incompleteness to be proven (or even described) then the numbers are
precisely this which is included in that subset of the theory.
Number Theory is the one with the largest (i.e., and infinite number) of
unprovable theories about numbers of the variations that I have been
considering. My point in the just prior post is that numbers are
precisely that item which the theory you are using to describe them says
they are, since they are artifacts created for computational
convenience, as opposed to direct sensory experiences of the universe.
As such, it doesn't make sense to say that a subset of number theory
leaves more facts about numbers undefined. In the subsets those aren't
facts about numbers.
Abram Demski wrote:
Charles,
OK, but if you argue in that manner, then your original point is a
little strange, doesn't it? Why worry about Godelian incompleteness if
you think incompleteness is just fine?
"Therefore, I would assert that it isn't that it leaves "*even more*
about numbers left undefined", but that those characteristics aren't
in such a case properties of numbers. Merely of the simplifications
an abstractions made to ease computation."
In this language, what I'm saying is that it is important to examine
the "simplifications and abstractions", and discover how they work, so
that we can "ease computation" in our implementations.
--Abram
On Thu, Oct 30, 2008 at 7:58 PM, Charles Hixson
<[EMAIL PROTECTED]> wrote:
If you were talking about something actual, then you would have a valid
point. Numbers, though, only exist in so far as they exist in the theory
that you are using to define them. E.g., if I were to claim that no number
larger than the power-set of energy states within the universe were valid,
it would not be disprovable. That would immediately mean that only finite
numbers were valid.
P.S.: Just because you have a rule that could generate a particular number
given a larger than possible number of steps doesn't mean that it is a valid
number, as you can't actually ever generate it. I suspect that infinity is
primarily a computational convenience. But one shouldn't mistake the fact
that it's very convenient for meaning that it's true. Or, given Occam's
Razor, should one? But Occam's Razor only detects provisional truths, not
actual ones.
If you're going to be constructive, then you must restrict yourself to
finitely many steps, each composed of finitely complex reasoning. And this
means that you must give up both infinite numbers and irrational numbers.
To do otherwise means assuming that you can make infinitely precise
measurements (which would, at any rate, allow irrational numbers back in).
Therefore, I would assert that it isn't that it leaves "*even more* about
numbers left undefined", but that those characteristics aren't in such a
case properties of numbers. Merely of the simplifications an abstractions
made to ease computation.
-------------------------------------------
agi
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