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

On 28 Jun., 02:58, "Dik T. Winter" <Dik.Win...@xxxxxx> wrote:
In article <1182860489.761901.119...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> WM <mueck...@xxxxxxxxxxxxxxxxx> writes:
> On 26 Jun., 04:38, "Dik T. Winter" <Dik.Win...@xxxxxx> wrote:
...
> > > I call it improper limit. Your colleagues from CVI have even not that
> > > much scruple and call it plainly limit.
> > >http://eom.springer.de/L/l058820.htm
> >
> > But *they* give definitions about what is meant here. Note the
> > *definitions* of epsilon neighbourhoods of oo (or -oo, or +oo).
>
> I know and use what they say, as every mathematician should know.

No, you did *not* use the definitions they give.

> The
> result can be cast into the plain form: If the sequence of reciprocals
> conerges to 0, then the (improper) limit of the sequence is infinity.
> And therefore your definition is wrong!

No. My definition is wrong in the context of *their* definition. But
that is *not* a definition about convergence within the real numbers.
You always come up with other definitions that contradict my definitions.
That is clearly possible. But my definition is *not* in the context of
those other definitions.

> > lim{n -> oo} a_n = A if for every eps > 0 there can be found
> > an n0 such that for n > n0, |A - a_n| < eps
> > does not work. It has changed to:
> > lim{n -> oo} a_n = A if for every eps > 0 there can be found
> > an n0 such that for n > n0, a_n is in an eps-neighbourhood of A.
> > Try to start reading stuff that you do understand.
>
> And if possible forgetting what you do not understand?

Sorry, I do not understand... Given the first definition of limit above,
now try to work out that:
lim{n -> oo} n = oo.
That is the definition of limits from analysis on R. The second definition
is the definition of topology on R u {-oo, +oo}. That you fail to see that
the two are different tells a lot.

> > What is your problem here? Sum{n in N} n cannot be defined in R using
> > conventional methods. Or do you know a conventional method such that
> > it can be defined? The only conventional method I know is limits.
> >
> This method yields the result oo. It is very conventional.

Perhaps, but oo is not in R. Or is it your opinion that oo is a real
number?

> But, of
> course, if you make an error you will always say the item is not
> conventionally defined.

No. What I state is that what is valid in one context is not
necessarily valid in another context. You are confusing definitions
from analysis with definitions from topology and defintions from set
theory. You are assuming that they should be the same, but they simply
*can't* be the same.

> > > Yes, it is the same case in the binary tree. Mathematics yields as
> > > many paths as nodes, musmatics yields Mus.
> >
> > Yes, you have an extremely limited view. Throw away Euler as a good
> > mathematician. He did similar to what I did.
>
> But e did not take it more seriously than his result 1 < -1.

How do you know?

> > > I know the one and only correct result of the sum in question in
> > > mathematics.
> >
> > Oh. According to your references I come to alpeh_0, oo, omega.
>
> Which is the same. Cantor exchanged oo by omega in the 80's and
> introduced alephs in the 90's.

Yes, using references over 100 year old. No, they are not the same.

They were invented, over 100 years ago (so my references must be that
old), as the same and they are the same. Only in order to veil your
errors and those of set theory you now try to make them different.

oo is used in analysis as a conventional symbol without any additional
meaning. aleph_0 and omega can be seen as being the same in set theory,
but *not* when you apply arithemetic.

> > > assertion 1/1 < 1/0 < 1/-1.
> > >
> > > Pardon, I forgot, the last one will be correct in your eyes.
> >
> > No, I see in the middle only something that is *undefined* (a concept
> > you apparently do not understand).
>
> Euler used and wrote the "number" oo and the number 2*oo and so on. A
> concept that you do not understand, but which is better justified than
> your sum = 0.

My definition does not need justification. As it is undefined (in the
context where I wrote), I can assign any definition I wish to it.

You are in error to think that something which is not well defined can
be assigned every value.

The sum 1-1+1-1+1-1+-... is not defined in the sense that is not a
real number, but it is well enough defined to exclude that the sum is
larger than 2.

> > So I reserve judgement. But it
> > looks closely to the one-point compactification of the reals.
>
> No, 1/1 < 1/0 < 1/-1 is simple nonsense. And every mathematician of
> today should recognize it as such.

What *is* true is that you cannot define ordering (assuming the ordering
axioms) on the one-point compactification of the reals.

My answer is that I call oo an improper limit. But the CWI (thanks for
explanation) does not care. In analysis.

Regards, WM


.

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