Re: abundance of irrationals!)

In article <1115571949.300178.29640@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Virgil wrote:
>
>
> > > > > > In mathematics, we never get "to infinity".
> > > > > > When we observe that something is taking
> > > > > > increasingly large finite values, that's the end of the
> story.
> > > > >
> > > > > But when we observe that Card({2,4,6,...,2n}) is always less
> than
> > > 2n,
> > > > > such that there are always numbers x (increasingly many with
> > > growing
> > > > > cardinality of the set) for which x/Card(set) --> oo,
> > > >
> > > > There is no logical relation between this statement
> > > > and the one below.
> > >
> > > Of course it is. Both are valid for All FINITE n.
> >
> > The expression x/Card(set) does not contain any 'n' so that
> "validity"
> > for any 'n' is impossible to determine.
>
> Read above: there are always numbers x, of course x e set =
> {2,4,6,...,2n}

If what you mean is " x/Card(S) for S = {2,4,6,...,2*n} and x in S, you
should say so clearly, instead of being deliberately vague and ambiguous.
>
> > > But
> > > that is not true. {1,2,3,4} is a finite set, but in {1,2,3,...,n} n
> can
> > > take on any value and all of its followers, as much as there may
> be.
> >
> > But each such set can be specified by a natural number, so it is that
>
> > natural number, not the set, which is critical to the induction
> process.
>
> Each natural number is finite. Each specifies such a set = sequence and
> vice versa. There is no number which was not speciefied by such a
> sequence. And how many such numbers ever are taken and considered
> together, there is nothing else resulting than the union of all of
> their sequences. Therefore any set of natural numbers is a sequence.
> But there is no last number, because for every sequence there is a
> larger sequence. The union of all sequences is N. Hence, any proof by
> induction concerning a sequence is valid for N.

False!
What is true for 1 and if for n then n+1 is true for all MEMBERS of N
but not necessarily for the set N itself.

According to WM's misinterpretation, N would have to be a finite natural
number, but it is neither finite nor a natural number.
>
> Regards, WM
.

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