Re: abundance of irrationals!)

In article <1115397094.324751.132770@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> *** T. Winter wrote:
> > In article <1115216969.144665.218720@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
> mueckenh@xxxxxxxxxxxxxxxxx writes:
> > >
> > > 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.

The actual infiniteness of at least one set follows directly and
specifically from the 'axiom of infinity' of either the Zermelo-Frankel
or Von Nuemann-Bernays-Goedel systems of axioms and Cantor's definition
of infinteness of a set.

Ant there is nothing "self-contradictive" about any of that.


> Which axiom results in a whole number aleph_0 which is lager than any
> natural?

What has beer to do with it?
>
> Regards, WM
.

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