Re: abundance of irrationals

From: W. Mueckenheim (mueckenh_at_rz.fh-augsburg.de)
Date: 02/22/05 

Date: 22 Feb 2005 13:25:45 -0800

David Kastrup <dak@gnu.org> wrote in message news:<x5psyuaya4.fsf@lola.goethe.zz>...
> mueckenh@rz.fh-augsburg.de (W. Mueckenheim) writes:
>
> > David Kastrup <dak@gnu.org> wrote in message
> >
> >> Well, he is also establishing some sort of informal bijection
> >> between sets and numbers and is using that to prove something.
> >> This is actually possible to do. The problem is that N is not in
> >> the sets that he places into bijection with the naturals. But
> >> since N is the _union_ of all those sets, you can prove properties
> >> about all members of N by proving them for all sets in the
> >> bijection with the naturals. That is as far as one can go. And WM
> >> then uses [...]to cover up that he claims to prove something about
> >> N itself (and not merely its elements).
> >>
> >> And that's where his reasoning breaks down. It is not a completely
> >> simple category mistake since there _is_ a bridge between naturals
> >> and sets. But this bridge does not extend to the union of all
> >> those sets itself, but only to its members.

That is your assertion. Can you name the slightest reason why it
should be correct?
> >
> > What is the difference between N and {n}, the ensemle, accumulation,
> > class, sequence, whatever you like to call it, of all finite
> > numbers?
>
> N is the set of all finite numbers. And it is an infinite set. So
> when you are establishing a bijection between finite numbers and
> finite sets, this bijection does not extend to N itself, even though N
> is the union of all such finite sets.

That is not an explanation at all but only a false assumption.
Induction covers all natural numbers.

> > What is contained in N which was not a finite number?
>
> Nothing, nobody ever claimed any different [...].

Therefore you should admit that your reasoning above is magic spell at
best.
>
> > I am sure you cannot an will not answer that question because you'd
> > have to admit that you have been talking nonsense.

> This question has been answered literally hundreds of times to you.

These "answers" including your "explanation" above are completely
nonsense. There is no reason at all to conclude (from the observation
that N has no largest element) that N was infinite. Only very
restricted minds must adhere to this primitive idea, after my
explanation how a set can be both simultaneoulsy finite and without a
largest element. Ok, there are such atavistic brains, but a human with
average mindpower should be able to understand my arguments.

Don't you see that there is not the slightest argument in favour of
your position?

Regards, WM

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