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I think that's tripping some people up is treating "infinity" as though it were a discrete number that can be compared. In discrete calculus, we were always careful to say a sequence could "approach" infinity faster than another sequence, which is both valid and fairly easy to understand.

The moment you treat "infinity" like a discrete number and manipulate it with discrete operators like comparison, you break the concept of infinity. Nothing is bigger than infinity. Infinity plus one isn't a discrete number, neither is infinity minus one. Sequences do not "equal" infinity because the transient property of equality would imply that all sequences approaching infinity are equal, which they're not.

One might be tempted to say infinity minus infinity is zero, but that's not semantically valid because infinity isn't a discrete number.

S1=1+2+3+4...

S2=2*S1

Both sequences are infinite, but neither are equal, nor do they "equal infinity". S2-S1 doesn't equal zero, it equals S1.

**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.

Well, first of all, there are two different concepts of infinity that had been discussed in some of this threads. One is the idea of infinity as a number that is larger than all the real numbers. The other is the idea of a set having an infinite number of elements. They are *not* the same concept. Let me give a brief description of both to see the difference:

1.- For certain applications, it is convenient to extend the real numbers to allow the values infinity and -infinity. (For example, this is a very common thing to do in measure theory and integration.) In this case infinity is (in a sense) a quantity that is bigger than all others quantities. The drawback of this approach is that infinity doesn't behave well under the usual arithmetic operations. For example infinity+1=infinity, while infinity-infinity is always undefined. In this setting you can't have anything bigger than infinity. Trying to manipulate divergent sequences as if they were the quantity infinity and doing algebraic manipulations with them is incorrect, and can quickly get you to absurd statements. For example, Riemann showed that if you have a conditionally convergent sequence, then you can rearrange its terms to make it have any limit you want.

2.- In set theory there is the notion of the "cardinality" of a set, which is a measure of how big it is. Two sets are said to be of the same cardinality if you can give a bijection between them. Of course, given any set you can always add one element to it to make it bigger. (This is based on the fact that there is no such a thing as "the set of all sets".) But having a set with more elements doesn't automatically means that it has a higher cardinality. For example the set of positive integers has the same cardinality as the set of even positive integers. (You just enumerate your positive even integers to get the bijection.) However, Cantor showed that the set of real numbers *does* have a bigger cardinality than the set of integers. This is very shocking. It means that it doesn't matter how you try to enumerate the real numbers, at the end there is always an infinite number of real numbers that you left out. Given any set, the set of all its subsets always has a higher cardinality, so there is no biggest cardinal number. (Again this is related to the fact that there is no "set of all sets".)

Member since:

2012-10-25

“So infinity +1 has no additional meaning.”

If the infinities in both cases are equal, then yes it does. Different size infinites are one of the fundamental principals of calculus. It's also one of the most difficult to understand.

Don't think in terms of sets where each number can only be used once. That's a purely artificial restriction that has absolutely no meaning whatsoever in this context. If you have a set of all reals, for example, you have a set with infinite members, none of which has a value of infinity. We are actually talking about infinite series. Not sets.

Lets say we have the following summation,

S1 = 1+2+3+4 + · · ·

We can all agree that the above goes to infinity. Now lets compare it with the following series,

S2 = 2+4+6+8 + · · ·

so that each number in the series is 2*n. Both are infinite. However, S2/S1 is exactly 2.

Because each term in S2 is divisible by 2, it is absolutely correct to write,

S2 = 2*(1+2+3+4 + · · ·)

In S2/S1 the series cancel out, their value (infinite or otherwise) is completely irrelevant.

S2, despite being infinite, is *exactly* twice S1, which is also infinite.

Whether a number has already been used is irrelevant. It's a sum. I can add the same value any number of times.

1+1+2+3+4 + · · · > 1+2+3+4 + · · ·