Can we discuss this?

[klaus54]

So according to you I think the operator + is a number.

Prove to me that ∞ cannot rationally be used in the same way as a number, like 0 (zero).

?

I don't have to "prove" anything to you. Infinity (sideways 8) is a symbol representing no upper limit to a function or sequence.

Aleph_null is a (cardinal) number representing "how many" integers there are. It's the first order of infinity (or first transfinite cardinal). And you can do arithmetic with transfinite caradinals. E.g., aleph_null + aleph null = aleph_null, aleph_null + aleph_null + aleph_1 = aleph_1.

Arithmetic with "sideways 8" makes no sense.

K54

P.S. 1) I am NOT a troll and 2) why are you arguing so vehemently about things you clearly don't understand? In this thread "infinity" has been explained over and over and over again.

If you want to do some research on transfinite cardinals, look up Cantor's diagonal proof and the Continuum Hypothesis.

Oh, BTW the notion of infinite sets is an axiom of Set Theory.

[Paprika]

You haven't actually defined it for all Real divisors, though. You've simply kicked the can down the road, a bit.

So, if you say that the division operator over a divisor of 0 is equivalent to this limit, you still have not defined it.

Quite. But then again, my aim wasn't to define the value, but to define the operation. From yours and Leonhard's replies I gather that my proposal to do so is quite valid, even though it can't achieve everything we might wish of it.

[Boxing Pythagoras]

Quite. But then again, my aim wasn't to define the value, but to define the operation. From yours and Leonhard's replies I gather that my proposal to do so is quite valid, even though it can't achieve everything we might wish of it.

Again, you haven't defined the operation. You've said that the operation is equivalent to another operation-- that's not a definition, that's a synonym. If two operations are equivalent, and one is undefined, the other is undefined, as well. And is undefined.

[Paprika]

is undefined.

I've pointed out repeatedly the distinction between the operation being defined and the resultant value being defined.

[Juvenal]

I've pointed out repeatedly the distinction between the operation being defined and the resultant value being defined.

This is incoherent. A binary operation isn't defined on a set if it's not closed on that set.

[Boxing Pythagoras]

I've pointed out repeatedly the distinction between the operation being defined and the resultant value being defined.

As lao tzu has stated, this doesn't make any sense.

It's as if you wanted to discuss a "schafloogle," and when I asked you what that was, you replied that it's a "garfloggle." You haven't actually defined anything. You've only given a synonym which is also meaningless.

[Paprika]

This is incoherent. A binary operation isn't defined on a set if it's not closed on that set.

Thank you. You certainly took your time getting here!

[pancreasman]

I believe that a duck can actually be used as a number and can give rise to meaningful mathematical equations. Prove I'm wrong.

[Juvenal]

I believe that a duck can actually be used as a number and can give rise to meaningful mathematical equations. Prove I'm wrong.

You are quite correct. Consider, for example, the "quack" operator, defined as follows ...

s quack t = t

Note immediately that, by definition:

s quack duck = duck

And further, that:

s quack t = duck -> t = duck

Providing algebraic rigor to the age-old truism:

If it quacks like a duck, it's a duck.

[klaus54]

You are quite correct. Consider, for example, the "quack" operator, defined as follows ...

s quack t = t

Note immediately that, by definition:

s quack duck = duck

And further, that:

s quack t = duck -> t = duck

Providing algebraic rigor to the age-old truism:

If it quacks like a duck, it's a duck.

Funny!

But it's a great example of how which symbols are used for numbers (or operators) don't matter. Algebra ("the joining") deals with how symbols combine.

Thanks,

K54

[JonathanL]

You are quite correct. Consider, for example, the "quack" operator, defined as follows ...

s quack t = t

Note immediately that, by definition:

s quack duck = duck

And further, that:

s quack t = duck -> t = duck

Providing algebraic rigor to the age-old truism:

If it quacks like a duck, it's a duck.

I don't get it, but I bet it must be funny, so you get an amen.

[Leonhard]

Quite. But then again, my aim wasn't to define the value, but to define the operation.

I'm afraid this doesn't make much sense, but even if it did it still wouldn't to define the operation as valid, but not the value. That's simple not the case and I can show you how that could land you in trouble.

Take a peek at this link to another thread I started on that: http://www.theologyweb.com/campus/sh...n-t-make-sense

[Juvenal]

I don't get it, but I bet it must be funny, so you get an amen.

The given "quack" operator, though not by that name, is actually a standard example from abstract algebra of an operation which is associative without being commutative.

a quack (b quack c) = a quack c = c
(a quack b) quack c = b quack c = c

Whereas:

a quack b = b
b quack a = a

Matrix multiplication is a more common example, but by its brevity, this one is considered more "elegant."

[pancreasman]

The given "quack" operator, though not by that name, is actually a standard example from abstract algebra of an operation which is associative without being commutative.

a quack (b quack c) = a quack c = c
(a quack b) quack c = b quack c = c

Whereas:

a quack b = b
b quack a = a

Matrix multiplication is a more common example, but by its brevity, this one is considered more "elegant."

I have vague recollections of teaching ring theory and so on. Enjoyed it immensely but I think most of it is gone now from my brain.

[Paprika]

Take a peek at this link to another thread I started on that: http://www.theologyweb.com/campus/sh...n-t-make-sense

My proposal has a flaw (that lao pointed out), but it is not that one: your thread assumes that my proposed definition of dividing of zero would be an inverse of the multiplication operator, which it isn't.