Re: Anybody in US using ADA ? One silly idea..

["Chad R. Meiners" <crmeiners@hotmail.com>]

"AG" <ang@xtra.co.nz> wrote in message
news:W1L%9.68847$F63.1401723@news.xtra.co.nz...
>
> "Grein, Christoph" <christoph.grein@eurocopter.com> wrote in message
> news:mailman.10.1044267682.3911.comp.lang.ada@ada.eu.org...
> > > > Do you know, an integer is definitely a real number.
> > >
> > > You sure? How about a simple theorem that any two distinct
> > > real numbers have another real number between them? Either
> > > this theorem doesn't hold (and the whole math goes out
> > > the window) or the integers are most definitely not reals.
> > >
> > > Which will it be?
> >
> > Where is there the contradiction? Integers can be embedded in reals as a
> subset.
>
> Operations available for the type and how they behave.
> The theorem above was an example. Let's see:
>
> F(X, Y) = A
>
> such that for any given, finite X and Y where X < Y it
> produces A that satisfies the requirement that X < A < Y
>
> Trivial to do for real numbers. Impossible for integers.
> Which means that, at least in this respect, integers are
> not a subset of reals. To be more precise, you could
> argue that they are a subset or real *values* but that's
> not the same thing.
>
> You could also define an operation F that takes two
> integers and returns a generic real but that would also
> mean that integers do not posess a property critical
> to the definition of the real numbers.
>

Of course integers are a proper subset of reals.  You cannot simply invent a
new definition for subset ;)  You are letting language confuse the issue.
To reformulate your statements above

Does there exist a function F(X,Y) with the signature F:R x R -> R such that
for all X and Y there exists an A such that
X < A < Y?

Yes.

and for the integers

Does there exist a function F(X,Y) with the signature F:Z x Z -> Z such that
for all X and Y there exists an A such that
X < A < Y?

No.

but let's get tricky here and ask a similar question about integer numbers

Does there exist a function F(X,Y) with the signature F:Z x Z -> R such that
for all X and Y there exists an A such that
X < A < Y?

Yes.  But wait those numbers are integers how dare there exist a number
in-between them that belongs to the reals! ;)

A number by itself has very few properties.  Sets on the other hand have
most of the properties you care about ;)
Subsets do not necessarily have the same properties as their supersets.

-CRM