Re: Cantor Confusion

In article <45700723.3060406@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

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


Large enough is certainly not qualitatively different enough, infinity
is the location where two parallel lines are thought to meet each other,
and division by zero has been forbidden because it yields anything.

Division by zero in a field yeilds a contradiction.

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

On the contrary, it resides in the definition of multiplication in the
set of objects one is considering.

Given addition in some set of objects, S, the general definition of
subtraction in S is that for elements a,b,and c in S, a - b = c if and
only if c is the unique x in S for which a = b + x.

Similarly given multiplication in a set of objects, S, then for
elements a,b,and c in S, a/b = c if and only if c is the unique x in S
for which a = b*x.

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.







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".
.

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. ... The residues of the integers modulus a prime is always a finite field ... It is better to understand the real reason, ...
    (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: ... Infinity has nothing to do with it. ... In finite rings, both are irrelevant, but the issue of division by zero ... It is better to understand the real reason, ...
    (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. ... Bob Kolker ...
    (sci.math)