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

In article <1183441649.404036.29620@xxxxxxxxxxxxxxxxxxxxxxxxxxx> WM <mueckenh@xxxxxxxxxxxxxxxxx> writes:
On 3 Jul., 04:37, "*** T. Winter" <***.Win...@xxxxxx> wrote:
....
> The declaration of 0 as a natural number.

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

If you can't see the facts supporting this opinion (discovery of 0
much later than discovery of genuine natural numbers) then further
discussion is meaningless.

So in giving names in mathematics you should consider history, otherwise
you are wrong? Sorry, in mathematics a term defines just what is given
in its definition. Nothing more, nor less. And different people give
the same name to different things.

Yes, especially your abuse.

If you can't see the facts supporting my discovery (lim_[x --> oo]
P(x)/K(x) in the binary tree) then further discussion is meaningless.

Yes, it is meaningless because you do not see that that limit is *not*
the necessary value.

> 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.

The left hand side is an abbreviation of the right hand side.

Wrong. It is only an abbreviation of the right hand side when the right
hand side is defined.

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.

SUM {n = 1..oo} n is undefined as the result of throwing dice is
undefined in advance. Nevertheless the result cannot be negative. So
much logic thinking should be available.
If you can't see this fact, then further discussion is meaningless.

You are still wriggling. The sum is undefined using ordinary mathematics,
nevertheless it is possible to give a definition (so much for your
identity). Stating that something that is undefined can be compared
with something that is well-defined borders on nonsense.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.