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Sun 19 Sep, 2010 04:19 am

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.

The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(do) : F(Ou) . Ou = Fu'.

@contrex,

contrex wrote:

The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(do) : F(Ou) . Ou = Fu'.

Before discussing the merit or correctness of your point, I must ask you what does it have to do with the division of zero by zero.

You don't divide anything by zero and hope for anything better than garbage.

The thing which occasionally IS done is to take a LIMIT of some fraction in which both numerator and denominator go to zero as the limit is approached but in which the part of the denominator which goes to zero can be factored out.

You, not me, equated it to Russell's paradox. Anyhow, zero is not a number, it is a cipher. Many apparent puzzles to do with zero ignore this fact. In ordinary (real number) arithmetic, division by zero has no meaning, as there is no number which, multiplied by 0, gives a (a≠0).

@contrex,

contrex wrote:You, not me, equated it to Russell's paradox. Anyhow, zero is not a number, it is a cipher. Many apparent puzzles to do with zero ignore this fact. In ordinary (real number) arithmetic, division by zero has no meaning, as there is no number which, multiplied by 0, gives a (a≠0).

Russell's paradox? When did I mention it? You are confusing the Liar paradox, to which I referred, with Russell's paradox, which deals with the set of all sets that do not include themselves.

So now zero is no longer a number?

@gungasnake,

gungasnake wrote:You don't divide anything by zero and hope for anything better than garbage.

The thing which occasionally IS done is to take a LIMIT of some fraction in which both numerator and denominator go to zero as the limit is approached but in which the part of the denominator which goes to zero can be factored out.

The thing that occasionally IS done is precisely dividing by zero, which is a common reason why computer programs crash -- as thus a main concern in computer science.

Dividing by zero does not produce garbage: much worse, it causes mathematics -- and computer programs -- to collapse.

In mathematics zero has 2 different roles to play. 1. To provide the symbol for the empty set. 2. To serve as a place holder symbol in a positional number system.

It is not a "number".

@guigus,

Well, what yo´re saying its truly false and falsely true...

@contrex,

contrex wrote:In mathematics zero has 2 different roles to play. 1. To provide the symbol for the empty set. 2. To serve as a place holder symbol in a positional number system.

It is not a "number".

You are definitely not a mathematician. The representation of an empty set is a slashed circle, not a zero: zero represents how many elements an empty set has, not that empty set itself. And you are confusing the symbol with whatever it represents. I am talking about a number, whatever its representation is. And although the belonging of zero to the natural numbers is controversial, its belonging to the integer, rational and real (only algebraic or not), etc, numbers is uncontroversial. Finally, a number has much more functions than two. Indeed, the very concept of a number, as given by Bertrand Russell, has nothing to do with a positional system or even counting:

A number is a class of similar classes.

Two similar classes have a one-to-one relationship between their elements, hence the same number of them.

@Fil Albuquerque,

Fil Albuquerque wrote:Well, what yo´re saying its truly false and falsely true...

Being true is neither being truly false nor being falsely true -- although any falsehood is also a truth as a falsehood. Above all, it's being truly true.

@guigus,

I never disagree with that...since a false truth is only false and a truthful falsity its only true.

@fresco,

As long I can keep up with the damn thing...

In the long term I am afraid A.I. will kill the fun...

I often wonder how Kasparov felt when "Deep Blue" kicked he´s ass...

@Fil Albuquerque,

Fil Albuquerque wrote:
As long I can keep up with the damn thing...

In the long term I am afraid A.I. will kill the fun...

I often wonder how Kasparov felt when "Deep Blue" kicked he´s ass...

Intelligence is intelligence, whether it is artificial or not, so I don't bother.

@fresco,

fresco wrote:
Ain't logic fun !

The only better thing I know -- besides sex -- is philosophy.

@Fil Albuquerque,

Fil Albuquerque wrote:I never disagree with that...since a false truth is only false and a truthful falsity its only true.

You are a bit confused: both a false truth and a truthful falsehood (not falsity) are falsehoods, despite being also true as falsehoods.

x/y =df (the z: x = y*z).

0/0 = (the z: 0 = 0*z).

But, 0 = 0*z is true for all z. That is, 0/0 is not unique, therefore 0/0 does not exist...even though it is defined.

Similarly, 1/0 does not exist because there is no z such that 1 = 0*z.

1/0 = (the z: 1 = 0*z).

x/0 does not exist for all x.

@Owen phil,

Owen phil wrote:

x/y =df (the z: x = y*z).

0/0 = (the z: 0 = 0*z).

But, 0 = 0*z is true for all z. That is, 0/0 is not unique, therefore 0/0 does not exist...even though it is defined.

Similarly, 1/0 does not exist because there is no z such that 1 = 0*z.

1/0 = (the z: 1 = 0*z).

x/0 does not exist for all x.

Plain and simple: zero divided by zero is any number, which is why it is said to be undefined. Even more simple (instead of more complicated, got it?): division is a series of subtractions. So 6/2 means how many times I can subtract 2 from 6, which is 3 times. Then, 0 / 0 means how many times I can subtract zero from zero, which is 1, 5, 96, or whatever number of times I wish. That is, any number of times. That's the meaning of the term undefined: zero divided by zero is any number I wish, which makes any number the same as any other. Now it is another thing entirely to say that 0 / 0 "does not exist:" on the contrary, the quotient of 0 / 0 exists, precisely, as any result I wish.

@Owen phil,

Owen phil wrote:x/y =df (the z: x = y*z).

0/0 = (the z: 0 = 0*z).

But, 0 = 0*z is true for all z. That is, 0/0 is not unique, therefore 0/0 does not exist...even though it is defined.

Similarly, 1/0 does not exist because there is no z such that 1 = 0*z.

1/0 = (the z: 1 = 0*z).

x/0 does not exist for all x.

Actually, the word mathematicians use for zero divided by zero is "indeterminate." The word "undefined" is used for any other number divided by zero -- for instance, one divided by zero. The word "undefined" means you cannot even conceive of something, while "indeterminate" means it can be anything. The division of zero by zero is different from any other division by zero, since:

1)

**No** number multiplied by zero equals a number that is

**different** from zero (undefined).

2)

*Any* number multiplied by zero equals a number that is

**identical** to zero (indeterminate).

So the correct word for zero divided by zero is not "undefined," but rather "indeterminate." Then, you could say that the quotient of one divided by zero does not exist, but you cannot say the same of the quotient of zero divided by zero -- the problem of zero divided by zero is not a problem of nonexistence, but rather that of an overwhelming existence -- the quotient of zero divided by zero exists: it is 1, 5, 96, or any other number.