Re: Irrelevant Relative Cardinality

In article <1121246758.347036.132280@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> 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.

There is a standard model of the reals based on Cahuchy sequences of
rationals that does not in any way depend on Dedekind cuts. There are
formal proofs of the complete equivalence of this model to the Dedekind
model, so the Dedekind model is not essential to the existence of the
reals. Learn some math before setting up as a critic of it, WM.
> > >
> > >
> > > > 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.

WM claims all sorts of proofs, but never provides any.

In mathematics, repeated claims, such as WM's backed only by claims of
proof but never backed up by anything that actually qualifies as
mathematical proofs are cast into outer darkness.

Which is the proper place for the irrelevance called relative
cardinality
.

Relevant Pages

  • Re: Relative Cardinality
    ... >> the axioms of an ordered field, which you should know and which do not ... > reals from the rationals, to show that there actually exists something ... > Since finiteness comes in so many flavors, ...
    (sci.math)
  • Re: Relative Cardinality
    ... the first 10^100 digits cannot be expressed together. ... The axioms for a complete Archimedean ordered field do not require ... reals from the rationals, to show that there actually exists something ...
    (sci.math)
  • Re: Interesting (IMO) question about the reals...
    ... Dedekind cuts, one can divide statements about the reals into two ... referred to in the real number axioms, such as addition of two reals ... would make sense to ask about _any_ complete ordered field. ...
    (sci.math)
  • Re: How many real numbers are there?
    ... the elements of my set T satisfy the ZF axioms. ... THEREFORE the cardinality of the reals is countable!! ... The construction of the reals (ie, a complete ordered field) does ... of elements of T, and each element of T is a set in ZFC, and therefore ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... >> through recursive calls to finite strings and the ZF axioms), ... >> is clearly a subclass of any other class of any model of the reals, ... I thought the word "first order formula" ... ONE has proven theorems about them (since I haven't seen new axioms ...
    (sci.math)
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