On Apr 14, 12:52 pm, "Dmitry A. Kazakov" <mail...@[EMAIL PROTECTED]
>
wrote:
> I think it is wrong to consider N and universal_integer equivalent.
Sure. It's {\bb Z} and universal_integer that are equivalent.
Seriously, we just disagree about this. I can't take
universal_integer seriously as a root class, because it's impossible
to write down any representation of it. I believe the approach I've
been thinking about could provide some reasonably solid grounding for
what universal integer is.
> Subseting is not a sufficient condition for a
> successful modeling.
In a discussion that's got a lot of formal logic in it, the word
"model" already means something pretty specific, usually involving a
Tarski structure. On the other hand, the informal use of the word
"model", in this context, is basically beside the point, which is to
get a precise definition of a hypothetical universal real type,
amongst others. There are plenty of useful things that are not-quite-
real numbers, such as the one- and two-point compactifications of the
real line, but these are the same thing as real numbers. If you want
them, fine; just don't try to claim that they are same thing and use a
confusing name for them.
> Actually it is the opposite, a perfect subset cannot
> be a good model.
There was paper from the fifties (sorry, no reference handy), which
used Turing machines to compute a Dedekind cut. On input a rational
number, it returned one of the two symbols "<=x" or ">=x". (You
cannot compute exact trichotomy without solving the halting problem.)
In any language describing these machines, there's a least one (Kleene
minimization), so there's a unique representative of such a machine
for every computable real number, which means there's a subset
bijection. Addition, multiplication, and their inverses are defined
in terms of the underlying operand-machine (it's pretty easy coding,
actually). So there's pretty close to a perfect model of the real
numbers, whose only real limitation is that run times are horrendously
slow. But it's also completely exact, with no compromises but
execution speed and representation size of a machine.
The subset is every real number that's computable. About as good as
you can do with computers, I'd say.
> The best models of R aren't even close to subsets. For
> example intervals with rational bounds.
In the precise meaning of model, it's just not a model, because
there's no total ordering on such intervals, so the ordering axioms
are not satisfied. In an imprecise meaning, it's real numbers plus
some other concept, which is more than {\bb R}.
Eric
|
