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BeamishBoy,

"Infinity is a well-defined concept. It's so well defined that one can talk about (i) different kinds of infinity, and (ii) how one version of infinity is 'larger' than another."

I didn't say it wasn't a well defined concept. It's just that the concept is being abused when we treat infinity as a discrete number as though it could be compared.

"I know this stuff is confusing but it's been understood by mathematicians for over a century."

It's really not that confusing, if you attempt to solve a discrete number for "infinity", anyone else can conceive of a number a number which is factually higher, therefor the concept of infinity rules out the possibility of any discrete number equaling infinity.

What we can do is compare sums of series which approach infinity at different rates, like you did earlier. Discrete calculus (which as you noted is well understood) allows us to solve the rate for any given iteration of the sequence and confirm via inductive proofs that it continues infinitely.

There are an infinite number of unique sequences who's sums add up towards infinity given an infinite number of iterations. Because of the transitive properties of mathematical equality, one cannot claim any of these infinite sequence sums "equal infinity", for the simple reason that they are not equal to each other.

We might be careless and say a given sum "equals infinity" and still understand one another, but we don't actually mean it in the true mathematical sense.

This is what my earlier example was trying to illustrate. If infinity could be treated as a discrete number, then mathematically this would be sound:

A=infinity

B=infinity

B-A = 0

But in fact the earlier example showed that B-A=A.

To be sure, I am arguing semantics in the conversation as a whole and not just you. I don't want us to talk over each other, and I don't think we have all that much to disagree on, despite your bolded statement that "This is completely incorrect."

*Edited 2013-03-24 20:57 UTC*

Member since:

2010-10-27

This is completely incorrect.Infinity is a well-defined concept. It's so well defined that one can talk about (i) different kinds of infinity, and (ii) how one version of infinity is "larger" than another.

I addressed this in another post by pointing out that the number of elements contained in the set of real numbers is one kind of infinity, and how this is smaller than the number of elements contained in the set of all subsets of real numbers.

I know this stuff is confusing but it's been understood by mathematicians for over a century.