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July 20th, 2009, 07:16 AM   #1
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infinity

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Originally Posted by CRGreathouse
Quote:
 Originally Posted by fibonaccilover Here is a question, someone told me that there are different sizes of infinity. How can this be if infinity goes on forever?
Why would two things that go on forever be the same?

If you really want an answer, start a thread on the applied math / other forum. As a teaser: intuitively, it seems that there are more real numbers than integers. Between 3 and 4 there are 3.5, pi, exp(1.3), and so on. Cantor proved that this intuition is accurate, in the sense that there are 'more' real numbers than integers.

But there are many types of infinite numbers, not all of which can be compared. This is because they're used for different things. You can talk about the first, second, or third object, but it doesn't make sense to talk about the pi-th object. Infinite numbers obey the same distinction between ordinal and cardinal numbers. But there are yet more types, depending on what you're doing and how you want things to work.
What are these types of infinite numbers? Also, out of curiosity, could you just name a couple examples or cases in which a different type of infinite number would be used?

 July 20th, 2009, 07:47 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: infinity Set theory has a wide variety of infinite numbers. These come in two types: cardinals and ordinals. Cardinals can be thought of as counting the number of elements of a set. For example, denote by |S| the cardinality of the set S. Then |{1, 7}| = 2. If Z is the set of integers and R the set of real numbers, then |Z| = aleph_0 and |R| = 2^aleph_0 > aleph_0. (This inequality is called Cantor's theorem.) There are also the extended real numbers (similar to the infinity symbol in calculus, comes in two flavors: +infinity and -infinity), the projective real numbers (only one type of infinity, which can be thought of as 'the smallest projective real' as well as 'the largest projective real'). There are also the hyperreals, the surreals, and many others.
 July 21st, 2009, 02:46 AM #3 Senior Member   Joined: Nov 2008 Posts: 199 Thanks: 0 Re: infinity Two sets are defined as being the same size if it's possible to exhibit a bijection (one to one correspondence) between their elements. Also, for a set $X$ define $2^X=\{\text{ functions from X to \{0,1\}\}$. It's possible to show that $2^X$ is strictly greater than $X$ (an injection is possible but no bijection) for all sets $X$. So, if we start with an infinite set, such as the set of all natural numbers, we must end up with a bigger set. This bigger set gives us a bigger (cardinal) infinity. We can then do the same thing to the new set to obtain another infinity and so on. This is not the whole story by any means but it does provide a whole pile of different infinites. There are others too but they are more difficult to explain.
July 21st, 2009, 06:20 AM   #4
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Re: infinity

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 Originally Posted by pseudonym There are others too but they are more difficult to explain.
So who's going to explain epsilon-naught?

July 21st, 2009, 07:49 AM   #5
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Re: infinity

Quote:
 Originally Posted by CRGreathouse So who's going to explain epsilon-naught?
I nominate...wikipedia! http://en.wikipedia.org/wiki/Epsilon_nought

July 22nd, 2009, 07:29 PM   #6
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Re: infinity

Quote:
 Originally Posted by CRGreathouse Set theory has a wide variety of infinite numbers. These come in two types: cardinals and ordinals. Cardinals can be thought of as counting the number of elements of a set. For example, denote by |S| the cardinality of the set S. Then |{1, 7}| = 2. If Z is the set of integers and R the set of real numbers, then |Z| = aleph_0 and |R| = 2^aleph_0 > aleph_0. (This inequality is called Cantor's theorem.) There are also the extended real numbers (similar to the infinity symbol in calculus, comes in two flavors: +infinity and -infinity), the projective real numbers (only one type of infinity, which can be thought of as 'the smallest projective real' as well as 'the largest projective real'). There are also the hyperreals, the surreals, and many others.
wow...I am going to have to sit down and think about this some more...there are a lot of types of infinities!

July 22nd, 2009, 07:32 PM   #7
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Re: infinity

Quote:
Originally Posted by pseudonym
Quote:
 Originally Posted by CRGreathouse So who's going to explain epsilon-naught?
I nominate...wikipedia! http://en.wikipedia.org/wiki/Epsilon_nought
This had me lost at "transfinite numbers"! I am not even going to try to understand this article! At least not until I get to a higher level of math!

July 22nd, 2009, 07:42 PM   #8
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Re: infinity

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 Originally Posted by fibonaccilover This had me lost at "transfinite numbers"! I am not even going to try to understand this article! At least not until I get to a higher level of math!
Transfinite number is just a different (fancier) way of saying 'infinite number'. But that article is too difficult to be of use here. Let me try to be easier -- though it'll still make your head swim if you can follow.

Basically, let's say you understand the smallest infinite ordinal ? = {0, 1, 2, 3, ...}. You're allowed to write ? + 1, the 'next' ordinal after ?. Keep going in this way and you get ? + 2, ? + 3, ..., ? + ? = ? * 2. After that comes ? * 2 + 1, ? * 2 + 2, ..., ? * 3. You can keep going in like fashion: ? * 3, ? * 4, ..., ? * ? = ?^2.

Like that? OK, it's pretty clear that you can write a bunch of ordinals in like fashion: ?^k * a_k + ... + ?^3 * a_3 + ?^2 * a_2 + ? * a_1 + a_0. This is (a subset of) Cantor normal form. It's easy enough to compare two numbers in this form: just find the highest power of ? and take the one with the greater coefficient. If they're the same, move on to the next-highest and so on.

Of course if we're going to write ?, ?^2, and so forth you just know that ?^? is coming. And from there it's a small step to ?^(?^?). Using that ellipsis that you know and love, I'll write
?, ?^?, ?^(?^?), ..., ??.

July 26th, 2009, 07:43 PM   #9
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Re: infinity

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by fibonaccilover This had me lost at "transfinite numbers"! I am not even going to try to understand this article! At least not until I get to a higher level of math!
Transfinite number is just a different (fancier) way of saying 'infinite number'. But that article is too difficult to be of use here. Let me try to be easier -- though it'll still make your head swim if you can follow.

Basically, let's say you understand the smallest infinite ordinal ? = {0, 1, 2, 3, ...}. You're allowed to write ? + 1, the 'next' ordinal after ?. Keep going in this way and you get ? + 2, ? + 3, ..., ? + ? = ? * 2. After that comes ? * 2 + 1, ? * 2 + 2, ..., ? * 3. You can keep going in like fashion: ? * 3, ? * 4, ..., ? * ? = ?^2.

Like that? OK, it's pretty clear that you can write a bunch of ordinals in like fashion: ?^k * a_k + ... + ?^3 * a_3 + ?^2 * a_2 + ? * a_1 + a_0. This is (a subset of) Cantor normal form. It's easy enough to compare two numbers in this form: just find the highest power of ? and take the one with the greater coefficient. If they're the same, move on to the next-highest and so on.

Of course if we're going to write ?, ?^2, and so forth you just know that ?^? is coming. And from there it's a small step to ?^(?^?). Using that ellipsis that you know and love, I'll write
?, ?^?, ?^(?^?), ..., ??.
ok, that makes much more sense to me than the article! Thank you so much for clarifying this concept to me. There is a lot to think about when it comes to infinity!

July 26th, 2009, 08:07 PM   #10
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Re: infinity

Quote:
 Originally Posted by fibonaccilover ok, that makes much more sense to me than the article! Thank you so much for clarifying this concept to me.
Glad I was able to explain it well enough for you.

Quote:
 Originally Posted by fibonaccilover There is a lot to think about when it comes to infinity!
Yes. The key thing to keep in mind is that there is no one concept of infinity, but many loosely-related concepts of infinite numbers.

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