Re: Relative Cardinality



Virgil wrote:

> > The question is: How many real numbers are really there? according to
> > the axioms of an ordered field, which you should know and which do not
> > state anything about an infinite set, but which include Dedekind's
> > cut-axiom.
>
> The axioms for a complete Archimedean ordered field do not require
> Dedekind cuts at all. They are only used to construct a model of the
> reals from the rationals, to show that there actually exists something
> satisfying those axioms.

Why do you always answer on topics I did not discuss? Do you feel so
much pleasure when your reply contains a "not"? The field of real
numbers is completed by the Dedekind-axiom.
> >
> >
> > > Then they will have advanced backwards, since all number are ideas, and
> > > nothing more.
> >
> > Numbers *are* more. At least the ideas required by Cantor's list must
> > have digits.
>
> How does that make them anything other than ideas?

It makes them ideas with digits. I call that numbers.
> >
> >
> > > > I have. It is an idea.
> > >
> > > It is, in fact, a member of the set of real ideas, which forms the
> > > Archimedean complete ordered field of ideas.
> >
> > Order is the outstanding property of a real number.
>
> Relevance?

Relative cardinality!

> Since finiteness comes in so many flavors (infinitely many), why is WM
> so insistent that non-finiteness have only one flavor?

It is proved by my relative cardinality.

Regards, WM

.

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