Re: Cantor Confusion

On 12/5/2006 1:14 AM, Virgil wrote:
In article <457467D5.7020201@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 12/1/2006 8:55 PM, Virgil wrote:
In article <45700723.3060406@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 11/30/2006 1:39 PM, Bob Kolker wrote:

Division by zero in a field yeilds a contradiction.

Just this contradiction resides already in the notion of (actual)
infinity.


Division by zero in standard sets of numbers is not defined because
there is never a unique x in such sets of numbers for which a = 0*x.
Either no x works or more than one works.

Infinity has nothing to do with it.

A finite example:

The residues of the integers modulus a prime is always a finite field
under the usual addition and multiplication, so there is no
"infinity" involved, but division by zero in those fields is still
barred for the reason above, a = 0*x can never have a unique solution.

I do not feel limited in thinking to the indefinitely large. I likewise
consider the indefinitely small (infinitesimal).

In finite rings, both are irrelevant, but the issue of division by zero
is the same even in such rings. Those who try to drag in the infinite or
infinitesimal in discussing the division by zero issue, just do not
understand the issue.

Hopefully you can substantiate this pure suspicion.
Being an engineer, I vaguely recall that a Zahlring is something like a
loop. Let me fantasize: {i, i^2, i^3} Is this a ring?
So far I do indeed not understand why the issue of rings matters in case
of division by zero.


Isn't it better to understand why it is incorrect than simply to learn
it is forbidden?

Eckard Blumschein

It is better to understand the real reason (see above), but Eckard
doesn't seem to understand the real reason. It has nothing to do with
"infinity".

Not directly with the indefinitely large, yes.

Where does the "infinitely large" or "infinitesmially small" enter into
finite rings, such as the fields of integers modulo a prime?

I do not grasp your point. Mathematical closed loops (meshs) are of
course pathways of infinite length. Correspondingly stars (nodes) add
all branches to the indefinitely small (zero).


The division by zero question has the same answer, and for the same
reasons, in these rings as in infinite rings.

Why not?


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