Re: *** T. Winter says: Definition: sum{i in N} i = 0

On 6 Jun., 18:39, Franziska Neugebauer <Franziska-
Neugeba...@xxxxxxxxxxxxxxxxxxx> wrote:
WM wrote:
On 6 Jun., 14:49, Franziska Neugebauer <Franziska-
Neugeba...@xxxxxxxxxxxxxxxxxxx> wrote:
WM wrote:
In

http://groups.google.com/group/sci.math/msg/d1b689f6e9820755?dmode=so...



*** T. Winter claimed:

Definition: sum{i in N} i = 0.

In reply I showed that this is wrong by three different proofs.

You cannot proof a *definition* "wrong".

If you define: A triangle is a figure that has three corners, and a
triangle is a figure without any corner, then this definition is
wrong. The proof is easy.

1. Your definition of triangle*

A figure is called triangle* iff
1. it has three corners, and
2. it has zero corners.

is not identical with the common definition of triangle

Worse. It is selfcontradictory. Not even an empty set of such
triangles exists.

2. The definitions itself are not considered statements. Hence they
itself do not carry a truth value. They are purely "syntactical" rules
in order to replace the words which are defined by what they are
defined.

That may apply to some definitions. It does not apply to the
definition a =/= a and something like that.

SUM{n>0}1/n is not less than 1.
Nicole d'Oresme proved this and even the unboundedless of the
harmonic series, i.e., the sum of unit fractions.

Even if it was the Pope your rise up against ***'s definition is of
no avail.

It was only a Bishop (of Lisieux). But what counts more: He was a
mathematician.

To experts your maneuver is known by the name "appeal to authority".
The problem is thathttp://pt.wikipedia.org/wiki/Nicole_d'Oresme
most likely never proved sum{i in N} i =/= 0 hence the reference
to him is futile.

I was not present when he (or not he?) proved the divergence of the
harmonic series. But I know this proof is correct.

But in its more general meaning, as positive infinity, it proves the
sum > 1.

1. How do you know that H&J do _not_ mean the "special meaning" of
aleph_0? The wish was father of the thought?

2. Can you give proof by quote that H&J meant (the naive) "positive
infinity" when they wrote "aleph_0"?

They prove that aleph_0 is larger than any natural number, in
particular they prove that aleph_0 is larger than 0.

4. To sum up, I can say that I am absolutely convinced that you _in_
_actuality_ do _not_ believe in their argument. Hence citing H&J is
pure diversionary tactic.

In actuality I believe that they are correct in proving the sum being
larger than 0. That's enough.

Regards, WM

.

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