Re: abundance of irrationals!)

In article <fb701d3c.0504180700.5696904f@xxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) wrote:

> imaginatorium@xxxxxxxxxxxxx wrote in message
> news:<1113650305.535670.12230@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>...
>
> > i.e. the SUM in a linearly ordered set.
> > > Form the (converging) SUM a_k over k = 1 to n.
> > > Do this for every finite n, without leaving out any.
> >
> > (How many of these finite n are there?)
>
> Some people say infinitely many. I would say: Their number cannot be
> quantified.

"Infinite" merely means "not finite", so it is quite correct to say that
the number of natural numbers is infinite (not finite).
> >
> > > This implies that one of the sums includes all n e N.

This assumption is false, and is one of the reasons why WM is not
competent to teach mathematics at anything above the most elementary
levels
> >
> > No it doesn't. For any n, I can show you its mother, sorry, I mean a
> > sum including it. I can't show you the single (finite) sum that
> > includes all of them, because it doesn't exist.
>
> For any line of Cantor's list, I can show you the next one.
> Nevertheless, Canor is believed to catch all of them.
> >
> > > If you disagree, tell me which n is left out.
> >
> > Oh, ok, try a different approach.
>
> No. Not ok, ok! You cannot name an n which always is left out. The
> only alternative is that all have been included.

Given a sequence of finite partial sums.
For each n, there is a sum for which the nth term is the last term.
This is obviously true for all n in N.

But WM claims it is not true for all n because there is a sum containing
ALL terms.

How can something which must be true for all terms not be true for all
terms?
>
> I'll give you another hint (with Z = set of whole numbers):
>
> 1) A n e N : E k e Z : k < n.
> 2) E k e Z : A n e N : k < n
>
> Both theorems are correct.

One example of reversing quantifiers without changing truth does not
prove it may be done universally.

For example, one replaces N with Z, the first remains true but the
second becomes false, so that the two quantifications are not equivalent.

Thus reordering existential and universal quantifications cannot be done
arbitrarily without changing meaning, and, at least sometimes, changing
the truth value of the quantified statement.
.

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