Re: Cantor Confusion

In article <4576CD31.2080808@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 12/5/2006 10:29 PM, Virgil wrote:
In article <457578DE.7030505@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

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.



Consider a set of 3 elements, say A = {x0,x1,x2} and binary operations +
and * , mappings from AxA to A that are commutative, associative, and so
that * distributes over + and such that
(1) for all a in A, x0 + a = a ,and x0 * a = x0 (x0 is a "zero" element)
(2) for all a in A, x1 * y = y (x1 is a unit element)
(3) other "additions" where a + b = b + a for all a,b in A
x1 + x1 = x2, x1+ x2 = x0, x2 + x2 = x1
(4) other "multiplications" where a*b = b * a fora all a,b in A.
x2 * x2 = x1

Alternately use the addition and multiplication tables below:

+ | x0 x1 x2 * | x0 x1 x2
---|--------- ---|----------
x0 | x0 x1 x2 x0 | x0 x0 x0
x1 | x1 x2 x0 x1 | x0 x1 x2
x2 | x2 x0 x1 x2 | x0 x2 x1

One may verify that { A, +, * } satisfies all of the properties of a
field.

One may define in it a subtraction "a - b" for a and b in A
by a - b = c if and only if Card({c:a = b + c, c in A } ) = 1,
i.e., a = b + c has one and only one solution

One may similarly define in it a division "a / b" for a and b in A
by a / b = c if and only if Card({c:a = b * c, c in A } ) = 1.
i.e., a = b * c has one and only one solution

According to this definition, division by x1 and x2 will always be
possible, but division by the zero element, x0, will not ever be
possible.

And the issue of "infiniteness" of a quotient is totally irrelevant.

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).

See the finite field example above. Division is not possible in it. But
also none of this infinite stuff is relevant in it either.

Electrical engineers like me benefit a lot from dualities and inversion.

it shows in your personality, dual and inverted.

In mathematics, subtractions are all defined in terms of unique
solutions to addition problems. Where no such unique solutions exist,
subtraction also does not exist.

In mathematics, divisions are all defined in terms of unique solutions
to multiplication problems. Where no such unique solutions exist,
subtraction also does not exist.

This is true in arbitrary rings, even those with no notions of size of
members. Thus smallness and largeness are irrelevant to whether a
division can be performed.

The division a/x can be performed if and only if there is a unique b
such that a = b*x. In a ring with more than 1 element, that cannot ever
happen.
.