Is infinity actually a variable or possibly an increasing number?

[The Student]

I am wondering if this makes sense.

Could infinitesimals be considered an infinitely decreasing number instead of a fixed number. Same for infinity, say aleph null, maybe this infinity is really just an increasing number instead of a fixed number. It would make sense for seemingly illogical statements like ∞ + 1 = ∞. And even though I only understand the very surface of the Banach–Tarski paradox
( http://en.wikipedia.org/wiki/Banach–Tarski_paradox ) it would seem to be able to explain how two spheres can come from one sphere of the same size.

Infinity also makes sense to me as a variable instead of a number. Let's say ∞ = x, where x has no identity and is ≥ ℕ elements.

 

[HallsofIvy]

There is no "numerical system" that considers "infinity" or "infinitesmals" as variables. It is not even clear to me what that would mean except to say that there are no "infinities" or "infinitesmals" in the number system. The value of a variables can be a number but variables are NOT numbers themselves.

 

[Quaid]

Could [an infinitesimal] be considered an infinitely decreasing number instead of a fixed number.

Same for infinity … maybe this infinity is really just an increasing number instead of a fixed number.

∞ + 1 = ∞

Infinity also makes sense to me as a variable instead of a number.

Did somebody teach you that an infinitesimal and infinity are each a fixed number?

Infinity is a concept, not a Real number. I think the same goes for infinitesimals.

A variable is a symbol that can represent multiple numbers (not simultaneously). When you write "variable instead of number" to describe infinity, there's an issue because infinity is not a Real number that can be assigned to a variable.

If you're working with the Real number system, you cannot do standard arithmetic with the infinity symbol. Infinity has no Real value; it's an idea.

What sort of math are you working on?

Cheers

 

[The Student]

Well, it was worth a shot.

 

[stapel]

Infinity is a concept, not a number.

Except, I think, in the "extended" real numbers.

 

[Quaid]

Well, it was worth a shot.

Sure! Glad to see that you're thinking about infinity.

 

[The Student]

Did somebody teach you that an infinitesimal and infinity are each a fixed number?

Infinity is a concept, not a number. I think the same goes for infinitesimals.

A variable is a symbol that can represent multiple numbers (not simultaneously). When you write "variable instead of number" to describe infinity, there's an issue because infinity is not a Real number that can be assigned to a variable.

If you're working with the Real number system, you cannot do standard arithmetic with the infinity symbol. Infinity has no Real value.

What sort of math are you working on?

Cheers

Oh I am just trying to make sense of all of this at a personal level. I am studying the early math with theory courses. It's tempting to try and jump ahead of what I am currently studying to try to tackle more interesting and more advanced problems. I know I shouldn't, but this time I just couldn't help it.

 

[Quaid]

Except, I think, in the "extended" real numbers.

Ah, I will insert the adjective "Real" in front of number.

That referenced page is interesting. They define the interval [−∞, +∞] using square brackets, and they say that neither −∞ nor +∞ represents a Real number.

When I think about infinity, it's usually in the context of some limiting process.

 

[Ishuda]

You might be interested in
http://en.wikipedia.org/wiki/Hyperreal_number
which, in some manner, connects infinity, infinitismals, and limits.