Re: Cantor Confusion

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.


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?

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

Relevant Pages

  • Re: Cantor Confusion
    ... Eckard Blumschein wrote: ... Infinity has nothing to do with it. ... In finite rings, both are irrelevant, but the issue of division by zero ...
    (sci.math)
  • Re: Cantor Confusion
    ... Eckard Blumschein wrote: ... Infinity has nothing to do with it. ... In finite rings, both are irrelevant, but the issue of division by zero ...
    (sci.math)
  • Re: Cantor Confusion
    ... Eckard Blumschein wrote: ... Large enough is certainly not qualitatively different enough, infinity ... Division by zero in a field yeilds a contradiction. ... It is better to understand the real reason, ...
    (sci.math)
  • Re: Weierstrass
    ... I would like to argue within the framework of set theory ... each one being included within a closed fence or container. ... between infinity and infinity plus or minus anything. ... Eckard Blumschein ...
    (sci.math)
  • Re: Cantor Confusion
    ... Eckard Blumschein wrote: ... Infinity has nothing to do with it. ... The residues of the integers modulus a prime is always a finite field ... It is better to understand the real reason, ...
    (sci.math)