Re: abundance of irrationals!)

In article <fb701d3c.0504191051.51f8665c@xxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) wrote:

> imaginatorium@xxxxxxxxxxxxx wrote in message
> news:<1113840031.725545.200050@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>...
>
> > It's not really an insult, is it? You really are being immensely dim.
> > (And this has nothing to do with "points of view": you can quite
> > happily say "I do not wish to discuss sets that are not finite", and
> > exit the discussion with grace.)
>
> I do wish to talk about sets that are infinite because it can be shown
> that they are not at all.

You have yet to show any such thing.
>
> I said: there is a sum over all n.
> You said: there is none.

We said that there is no definition of any such thing in mathematics
EXCEPT as the limit of the sequence of partial sums, provided that limit
exists. Since you seem to reject this definition, you reject such "sums".

> I said: which n is excluded.

>From what? From the set of finite sums, none are excluded, from any
infinite sum, all are excluded.

> You are incapable of naming on, but continue you insult.

it is not an insult to tell the truth.

They are all excluded if you do not accept the definition, as there is
nothing to include them in.

Finite sums are defined by induction on the number of terms to be added,
once two terms can be added, but this sort of induction does NOT extend
to infinite sums.

> A very dishonest behaviour.

That is our opinion of your behavior.
>
> > If you can't do that, we can assume there's something wrong with your
> > "logic".
>
> Not with mine, but with yours. See where your example fails:
>
> 1) A n e N : E k e Z : k < n.
> 2) E k e Z : A n e N : k < n

Do you claim that these two quatifications are always equivalent for
arbitrary ranges of the variables?

Consider
1) A n e N : E k e N : n < k. (TRUE)
2) E k e N : A n e N : n < k. (FALSE)

So there are times when the order of quantification makes a difference.

WM seems to think the order of quantification never makes any
difference, but one example where there is a difference proves him wrong.

The above is such an example.

>
> Regards, WM
.

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