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.
Abram Demski wrote:
Charles,
Interesting point-- but, all of these theories would be weaker then
the standard axioms, and so there would be *even more* about numbers
left undefined in them.
--Abram
On Tue, Oct 28, 2008 at 10:46 PM, Charles Hixson
<[EMAIL PROTECTED]> wrote:
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.
-------------------------------------------
agi
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