Dividing Zero By Zero

guigus, All I've got to say at this point is that you have the patience of a saint.

The good news is, I think I'm beginning to grasp a little of what you're trying to convey. Don't get your hopes up too much, but I want to thank you for taking the time with this old stubborn geezer whose gray matter has lost much of its learning ability.

I had to look up the definition of "biunivocal relations " for a clearer (to me) explanation:

"Biunivocal Relation is a mathematical term meaning one-to-one. A biunivocal relation between sets A and B means that an element a from set A is related to one and only one element b from the set B which, in it's own right, is solely related to a.

The difference between this and univocal is that univocal does not exclude the possibility of many-to-one relations, where for example, one word has a single meaning but its synonyms have the same meaning. A biunivocal relation would require words to have a single meaning as well as it would require each meaning to be ascribed to a single word (no synonyms!)"

If we ask whether dividing zero by zero equals one, then we must answer yes, because if we multiply one by zero then we recover zero. However, if we ask whether it equals two rather than one, then we must still answer yes, because if we multiply two by zero then we also recover zero. Likewise, the result of dividing six by two is three only because multiplying three by two results in six. Hence, the result of dividing zero by zero is any number, as multiplying any number by zero results in zero. So dividing zero by zero makes any number the same as any other number. And if all numbers are the same, then all operations between them are also the same. Yet still, in taking a single division of zero by zero with any single quotient, we must recognize it as a perfectly valid operation, since this quotient multiplied by zero results in zero. Nothing at each single division of zero by zero alone makes it invalid: each one is a place where all numbers falsify themselves. Conversely, these operations are only possible if any number is different from any other number, hence true, so every number must be true to be false -- in the division of zero by zero -- and false to be true -- since the division of zero by zero remains valid.

Likewise, whenever I say, "right now, I am lying," if what I am saying is true, then it must be false, and if it is false, then it must be true.

Don´t mind the healthy harassment of my remark, but is n´t there the quality of quantity ?
That much is there I figure...

guigus wrote:

If we ask whether dividing zero by zero equals one, then we must answer yes, because if we multiply one by zero then we recover zero. However, if we ask whether it equals two rather than one, then we must still answer yes, because if we multiply two by zero then we also recover zero. Likewise, the result of dividing six by two is three only because multiplying three by two results in six. Hence, the result of dividing zero by zero is any number, as multiplying any number by zero results in zero. So dividing zero by zero makes any number the same as any other number. And if all numbers are the same, then all operations between them are also the same. Yet still, in taking a single division of zero by zero with any single quotient, we must recognize it as a perfectly valid operation, since this quotient multiplied by zero results in zero. Nothing at each single division of zero by zero alone makes it invalid: each one is a place where all numbers falsify themselves. Conversely, these operations are only possible if any number is different from any other number, hence true, so every number must be true to be false -- in the division of zero by zero -- and false to be true -- since the division of zero by zero remains valid.

Likewise, whenever I say, "right now, I am lying," if what I am saying is true, then it must be false, and if it is false, then it must be true.

You are correct, if we define x/y as (a number z such that: x=y*z).
That is to say, 0/0 = (a number z such that: 0 = 0*z).
But (0*z = 0), is true for all numbers z.
Therefore,
0/0 = 0 and 0/0=1 and 0/0=2 etc..
This proceedure leads to the absurdities that you mention.

Logic/mathematics abhors contradictions..Bertrand Russell.

For example..
0/0 =df (a z: 0=0*z) -> 0=1.
~(0=1).
Therefore,
~(0/0 =df (a z: 0=0*z)).

You are wrong to define x/y =df (a z: x=y*z).
(Because it leads to contradictions.)

The correct definition of x/y, which does not lead to your absurdities is:
x/y =df (the z: x= y*z).

Your failure to distinguish between the indefinite article (a) and the
definite article (the), is the source of your errors here.

See: On Denoting..B. Russell.
http://www.cscs.umich.edu/~crshalizi/Russell/denoting/

D1. G(a z: Fz) =df Ez(Fz & Gz).
D2. G(the z: Fz) =df Ey(Az(z=y <-> Fz) & Gy).
D3. x/y =df (the z: x= y*z).

~(0/0=0).

Proof:

1. 0/0 = 0 <-> Ey(Az(z=y <-> 0=0*z) & y=0).
By: D3, D2.
2. 0/0=0 -> Ey((0=y <-> 0=0*0) & (1=y <-> 0=0*1)).
By: 1, AxFx -> (F0 & F1).
3. 0/0=0 -> Ey((0=y & 1=y).
By: 2, 0=0*0, 0=0*1.
4. ~Ey(0=y & 1=y). elementary theorem.
5. 0/0=0 -> contradiction.
By: 3, 4.
Therefore,
6. ~(0/0=0).
By (p -> contradiction) <-> ~p).
QED.

Similarly...~(0/0=1) and ~(0/0=2) and ~(0/0=3) etc..
That is to say, there is no number that is equal to 0/0.

~Ex(x=0/0) is a theorem of elementary arithmetic.

0/0 does not exist!

There is no thing that is equal to 0/0, including itself.
~(0/0=0/0), is tautologous.

Owen phil, Thanks for bringing me back to where I started on this merry-go-round. I always had this intuitive belief in 0/0=0, and the way you explained it confirms that belief.

I owe you some drinks.

Computers crash? LOL Now, there's a twist I would never have thought of. What if computers never existed? Does that math still apply?