What do you mean by believing in any kind of number? There's no more reason to believe in 4 than ∞.

 

And what do you mean by no useful quantities? Infinity maybe a quantity if of itself (the quantity of rational numbers is countably infinite etc.) but it doesn't mean anything for it to "have a quantity".

True. I believe then infinity sign = unknown.

Why? Infinity is pretty well defined in my book. I know exactly what it means, I know what it means in the context of limits, I know what it means in the context of cardinalities and I know what it means simply as a number. You can probably get all of that from just reading this thread.

 

How can such a thoroughly defined number be unknown?

well, if infinity is unknown then 0,1,2,3,4,5,6,7,8,9.... > infinity, must be equally unknown as they are equally or less defined.

Perhaps I am engaging in semantics, I understand your argument, but unless a problem can be calculated to infinitum what real purpose does is have?

As you really should have already gathered, uses for infinity are:

 

For calculating limits: [math]\sum_{n=1}^{\infty}\frac{1}{2^n}=2[/math]

 

And describing cardinalities: [math]|\mathbb{N}|=\aleph_0[/math] (the cardinality of the set of natural numbers is countably infinite).

 

And other things besides.

if there's a positive infinity, can there be a negative infinity?

 

Sure there is. Take the graph of y=1/x and approach 0 from the negative side.