Re: abundance of irrationals!)

In article <1115216969.144665.218720@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Virgil wrote:
>
> > >
> > > But there is no proof that this Card(N) is a meaningful notion at
> all.
> >
> > The definition of 'cardinality' is an equivalence class under the
> > equivalence relation of being in one to one correspondence (having a
> > bijection from one to the other).
> >
> > Thus any set has a well defined cardinality.
> >
> Obviously this is a meaningless and self-contradictive definition,
> which easily cshould be abandoned because it (and the actual existence
> of any infinite set) does not follow from any axiom.
>
> Regards, WM

The existence of a set satisfying Cantors definition of being infinite
is guaranteed in the Zermelo-Frankel axiom system by the axiom called
"the axiom of infinity".

http://mathworld.wolfram.com/AxiomofInfinity.html

The axiom of infinity is also one of the von Neumann-Bernays-Gödel
axioms.

That axiom is so well established in mathematics that closing your eyes
to it, as you do, will not make it go away.
.

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