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

In article <1183010974.179614.181900@xxxxxxxxxxxxxxxxxxxxxxxxxxx> WM <mueckenh@xxxxxxxxxxxxxxxxx> writes:
On 28 Jun., 02:35, "*** T. Winter" <***.Win...@xxxxxx> wrote:

> >
> > Depends on the definition of N you are using. Remember Bourbaki.
>
> Forget them as soon as possible.

Why? What is wrong with their books?

The declaration of 0 as a natural number.

Again, nothing more than opinion. Moreover, they are not the only ones
who do that.

BTW, it is
*not* CVI but CWI: "Centrum voor Wiskunde en Informatica", or "Centre
for Mathematics and Computer Science" in translation. And if you
wonder at the term "Wiskunde", that is one of the many Dutch scientific
terms used in Dutch as invented by Simon Steving, the man who brought
decimal fractions to you.

He would rotate in his tomb could he see the abuse of his famous
invention.

Yes, especially your abuse.

> > You are wrong. The identity holds *if and only if* the right hand
> > part exists. And in that case the identity is a *definition* for the
> > left hand side.
>
> We have different opinions. My opinion is: The identity is an identity
> any case, and your opinion is wrong.

In that case you assume identity between something undefined and something
well-defined. Good luck.

Your assertion is wrong, because an identity implies that both parts
are simultaneously defined or both are undefined.

SUM_[n = 1 to oo] a_n == LIM_[k --> oo] SUM_[n = 1 to k] a_n

But in that case there *must* be a definition of the left-hand side without
reference to the right hand side. But whatever, this make
sum{n = 1..oo} n
undefined because
lim{k -> oo} sum{n = 1..k} n
is undefined in ordinary mathematics. In H&J the "sum" above is *not*
defined using limits.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.