On Mon, 03 Dec 2001 17:29:39 GMT, "Paul Lutus" <email@example.com> wrote:
> "The Scarlet Manuka" <firstname.lastname@example.org> wrote in message > news://email@example.com... > > > I don't think that the extended reals would be a good default > > number system. You seem to insist that since I want them to be > > a number system, I want them to be the default. That is simply > > not true. > > You think there is no meaningful default definition for "number > system", a view not shared by many others, but you think if this > was feasible (on your own terms) than the extended reals should > be included in that definition.
There are defaults, plural, and they are context-dependent.
> As you put it: > > > I said: The term "number system" is > > not well-defined, and it would be > > unwise to define it in such a way as > > to exclude the extended real numbers. > > Surely you see that a specific definition for "number system" must > either include, or exclude, the extended reals. For reasons of > consistency, such a definition cannot include the extended reals > when that is convenient, and exclude them at other times. So the > expression "define it in such a way as to exclude the extended > real numbers" is, ipso facto, to exclude the extended reals > altogether, thus conflating with the default definition of > "number system" shared my most others, and conversely (if extended > reals are included) that the described system would become the > extended reals.
There is not one "number system". "Number system" is a class, one member of which is the extended system of real numbers.
If The Scarlet Manuka had wanted to be hyper-precise, he might instead have said something like:
The class "number system" is not well-defined, and it would be unwise to define it in such a way as to exclude the extended system of real numbers from membership.
You are imagining that there is one large "number system" (whether precisely defined or not) containing all the "numbers" belonging to all the "number systems" in the class. But nobody has been arguing for such a view.
> So to argue, as you have, that a proposal for a default definition > for "number system" should *include* the extended reals is equivalent > to saying that such a definition *would be* the extended reals.
This follows only from the false premise outlined above.
> > > This most recent exchange has been positively schizoid.
I'm not sure of the correct diagnosis. :)
> Again, the problem you face is that I understand what you are saying.
> There is no chimerical "number system" that somehow "includes" the > extended reals that is not by definition the extended reals (or an > extension of that, with all its properties plus others).
What there is is a rather vague understanding of what counts as *a* "number system". (Sort of a Wittgensteinian family resemblance, I guess.)
-- Angus Rodgers (angus_prune@ eats spam; reply to angusr@)