- #1

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How do infinities work in operations?

Are there "smaller" and "bigger" infinities?

If ∞+1=∞, is ∞-∞=1?

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- Thread starter f25274
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- #1

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How do infinities work in operations?

Are there "smaller" and "bigger" infinities?

If ∞+1=∞, is ∞-∞=1?

- #2

- 65

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Even then, however, we still can't define notions like "∞-∞". However, ∞+1 = ∞ makes perfect sense.

We can formalize it as follows:

x*∞ = ∞ for any x not equal to 0

∞+x = ∞ for any x not equal to ∞

∞-x = ∞ for any x not equal to ∞

x/0 = ∞ for any x not equal to 0

But notice that we can't apply the normal rules of algebra to equations anymore. For example, we cannot multiply both sides of the equation "x/0 = ∞" by 0, because we are not allowed to multiply ∞ by 0 (in much the same way we are not normally allowed to divide by zero!).

- #3

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- #4

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In this case, however, "infinity" is not considered a number, but a size. We use different notions of infinity to describe a notion of the size of a collection (or set) of objects. For example, there are infinitely many natural numbers.

When using infinity to describe the size of a collection of things, we call it "cardinality". Instead of saying "There are infinitely many rational numbers" we say, "The rational numbers are countable". "Countable" is a cardinality (i.e., a type of (infinite) size).

We say a collection of objects is countable when we can assign a unique natural number to each of those objects, such that every natural number is used exactly once.

Now, we can also say that a collection of objects has a specific finite cardinality. For example, the set of letters {r,b,g,f,m} has cardinality 5, because I can use ALL the numbers 1 through 5 to label them and do so uniquely. i.e.:

1. f

2. m

3. g

4. r

5. b

The rationals are countable, because we can make a similar list of all fractions:

1. 1/2

2. 35/29

3. -4/5

...

I won't go into how to prove that you can do that!

Anyway, it turns out that the cardinality of natural numbers is the "smallest" infinity there is. The next biggest is the cardinality of the real numbers.

- #5

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an infinite number of numbers but if it includes all integers you have ...,-3,-2,-1,0,1,2,3,... which includes infinite numbers but more than the first sequence. Am I right?

- #6

Hurkyl

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There are a variety of algebraic and geometric structures that have infinite elements.

How do infinities work in operations?

Are there "smaller" and "bigger" infinities?

If ∞+1=∞, is ∞-∞=1?

The progression of numbers you learned in school is quite misleading, because those number systems have unusually wide application, and they give the impression there's a "one true system" of numbers and we are only considering which ones to use.

The infinite numbers you use to measure

If you want to measure sequences in a way that ...,-3,-2,-1,0,1,2,3,... is bigger than 0,1,2,3,4,5,..., then cardinality is not appropriate for that purpose -- the sets of numbers in those sequences have the same cardinality ([itex]\aleph_0[/itex], the cardinality of the natural numbers), as can easily be seen by reordering them:

Code:

```
0, 1, 2, 3, 4, 5, 6, ...
0, -1, 1, -2, 2, -3, 3, ...
```

The two sequences you stated have different

Order type can't distinguish between

Code:

```
0, 1, 2, 3, 4, 5, 6, ...
1, 2, 3, 4, 5, 6, 7, ...
```

You could, of course, compare them by instead trying to measure the difference of the two sets.

- #7

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Let me clarify this:

I know there are more terms between 0 and 2 (counting fractions) than between 0 and 1.

They both have infinite terms between them but is the cardinality between 0 and 2 but is it twice the cardinality between 0 and 1?

I know there are more terms between 0 and 2 (counting fractions) than between 0 and 1.

They both have infinite terms between them but is the cardinality between 0 and 2 but is it twice the cardinality between 0 and 1?

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- #8

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They both have infinite terms between them but is the cardinality between 0 and 2 but is it twice the cardinality between 0 and 1?

Surprisingly, no. When comparing cardinalities of two sets A and B, we look for one-to-one correspondences between them, which is a way of identifying each element of A with a unique element in B. We say that two sets have the same cardinality if there is such a correspondence between them. I can easily make such a map between [0,1] and [0,2]: just multiply by two. It takes a little proving to show that this function is really one-to-one (that is, each element of [0,1] gets identified with a unique element of [0,2]) and onto (every element of [0,2] gets "hit" by something from [0,1]).

So even though intuitively, there are "more" numbers in [0,1] than in [0,2], these sets have the same cardinality. Infinite sets can lead to some very counter-intuitive results!

- #9

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...They both have infinite terms between them but is the cardinality between 0 and 2 but is it twice the cardinality between 0 and 1?

Surprisingly, no. [...]

So even though intuitively, there are "more" numbers in [0,1] than in [0,2], these sets have the same cardinality. Infinite sets can lead to some very counter-intuitive results!

Technically the answer to the question as posed is "Yes", because twice an infinite cardinal is equal to that cardinal value. This isn't so strange, after all, the cardinality of the empty set is also equal to twice its value as well.

You can define addition and multiplication (and powers) of cardinal numbers consistently as long as you avoid subtraction and division (which is simply the absence of negative or reciprocal cardinal numbers). There are corresponding set operations e.g. disjoint unions for +, cartesian products for *, and "sets of all mappings" for ^.

BTW The OP mentioned

Check out the wikipedia article: http://en.wikipedia.org/wiki/Transfinite_number" [Broken] for more details.

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- #10

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Technically the answer to the question as posed is "Yes", because twice an infinite cardinal is equal to that cardinal value. This isn't so strange, after all, the cardinality of the empty set is also equal to twice its value as well.

Oops, my mistake!

- #11

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For example:

Taking the integral of 1 to infinity of 1/x^2 does not exist as a Riemann Integral.

It has a domain that is unbounded.

If one tries to invoke the Second Fundamental Theorem and plug and chug, it works.

By conventional wisdom,

(-1/infinity) - (-1/1) = 1

However, if you were to write this on a test, it would not be given credit. -1 CANNOT be divided by infinity. Afterall, infinity contains many, many, many numbers. Is it 1/2 or 1/100000?

Instead, we take the integral from 1 to n. We then take the indefinite integral and plug in n.

This gives us -1/n - (-1/1).

To find your integral, we find the limit of the above as n approaches 0.

limit as n approaches 0 of (-1/n) - (-1/1) = 1.

Giving us the correct answer.

Hope that helps. I tried to keep my explanation limited to someone who has taken basic Calculus courses.

- #12

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That's a really long way of saying that infinity is not in the set of real numbers. :tongue:

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