Re: abundance of irrationals!)

In article <1115307636.877172.177000@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Franziska Neugebauer wrote:
> > Dear Wolfgang,
> >
> > mueckenh@xxxxxxxxxxxxxxxxx wrote:
> >
> > > Every non-empty set of natural numbers contains at least one number
> > > which is not smaller han the cardinality of that set. This theorem
> > > does not state anything about the cardinality of the set. It is
> proven
> > > for any finite set, because we dobt the existence of infinite sets.
> > > Should they exist, however, volation of that theorem would show
> them
> > > to not exist.
> >
> > ROTFL. What about this application of your reasoning:
> >
> > Theorem: Every apple is green.
> > It is proven only for green apples, because we doubt the
> existence
> > of red apples. Should red apples exist, however, a violation of
> that
> > theorem show them to not exist.
>
> Dear Franziska,
>
> this bijection
> n <--> Card({1,2,3,...,n})
> is defined for EVERY natural number n e N (not for the green numbers,
> but for the finite ones, because there is a rumor that others weren't
> available). Now there are precisely three (3) alternatives:
> 1) Either, you say N is a finite set (then the proof is invalid but not
> necessary at all)
> 2) or you can specify any n e N for which my bijection is undefined.
> 3) or you accept that the bijection covers infinite sets too.
> There is *no* (0) other possibility!

There is always the possibility that WM is wrong.

Your option (3) is ambiguous, it actually covers two situations:
(3a) there are infinitely many pairings n <--> Card({1,2,3,...,n}}) but
all of the n's and Card({1,2,3,...,n}})'s in those pairings are finite.
(3b) some n's and some Card({1,2,3,...,n}})'s become infinite.

You imply (3b) but actually (3a) is what happens.

But as you do not like that, you did to list it.
.

Relevant Pages

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