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Different kinds of infinities?

Today I got to know that two infinity can be compared, But I want to know how is this possible? infinity will be infinity. If it doesn't have any particular value, how can we say that this infinity is small and other one is greater. Can anyone help me?

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marked as duplicate by Jennifer Dylan, Brian M. Scott, Asaf Karagila, Henning Makholm, Austin Mohr Aug 21 '12 at 17:21

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possible duplicate of Different kinds of infinities? Also Are all infinities equal? –  user2468 Aug 21 '12 at 16:54
    
So what did you hear today? –  Thomas Aug 21 '12 at 16:54
    
@Thomas: that two infinite values can be compared and we can say which is greater and which one is smaller. can you prove it? –  Rahul Taneja Aug 21 '12 at 16:57
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I might be wrong here, but I don't think you can go far without set theory. Infinity here is defined as a size (cardinality) of a set. Take a look at ordinal numbers and cardinal numbers and The Book of Numbers by J. H. Conway & R. Guy. –  user2468 Aug 21 '12 at 17:09
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@RahulTaneja: If you want to, you can declare that any two infinite sets are equal in "size." No problem arises from this declaration. It just means that "size" and cardinality are different notions. Before Cantor, that was how things were done: there was no attempt to distinguish between different "infinities." But the distinctions introduced by Cantor have proved to be very useful in mathematics. –  André Nicolas Aug 21 '12 at 18:31
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The question about several types of infinity has come up several times before. As linked to in the comments about, you can read through the question and answer given in Different kinds of infinities? or Are all infinities equal?.

You ask about if you can compare infinities. Well, you might say that you can. IF you take for example the natural numbers $1,2,3,...$, then there are an infinite number of them. We say that the set is infinite. But, you can also count them. If we look at the real numbers, then the fact is that you cannot count these. So in a way, the infinite number of real number is "greater" than the infinite number of natural numbers.

But all this comes down to the question about how you measure the size of something. If someone says that something is bigger than something else, then they should always be able to define exactly what that means. We don't (I don't) like when questions become philosophical, then it has (In my opinion) left the realm of mathematics. So if someone tells you that one infinity is greater than another infinity, ask them exactly what they mean. How do you measure sizes of infinities? If they are a mathematician, they will be able to give you a precise definition (study Andre's answer).

But, what we usually think about when we compare numbers (or elements in a set) is a set with some kind of ordering on. Without going into any detail, there are different types or orderings, but you can think about how we can order the set consisting of the real numbers in the usual way (ex $7 > 3$). But in this example we are just talking about the real numbers. And infinity is not a number.

One more thing to keep in mind is that we will some times write that a limit is equal to infinity. Like $$\lim_{x \to a} f(x) = \infty. $$

However, when we write this, we don't think (I don't) of $\infty$ as an element in the set of real numbers (it isn't). All we mean by writing that the limit is infinity is that the values of $f(x)$ become arbitrarily large as $x$ "gets" close to $a$.

Just a few things.

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$+1$ for the part about the limits. I guess that causes quite a confusion when one first learns about set-theoretic infinity. –  user2468 Aug 21 '12 at 17:17
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Two sets $A$ and $B$ are said to have the same cardinality if there is a function $f: A \to B$ which is one-to-one and onto. More informally, $A$ and $B$ have the same cardinality if the elements of $A$ and $B$ can be "paired off." Note that if $A$ is finite, then $A$ and $B$ have the same cardinality if and only if $A$ and $B$ have the same number of elements. But the formal definition of "same cardinality" does not mention numbers, so it makes sense even for infinite sets.

Let's look at an infinite example. The set $A=\{1,2,3,4,\dots\}$ of positive integers and the set $B=\{2,4,6,8,\dots\}$ of even positive integers have the same cardinality, for we can pair off the integer $k$ with the even integer $2k$. In terms of functions, the function $f(x)=2x$ is a one-to-one onto mapping from $A$ to $B$.

We say that $A$ has cardinality less than (the cardinality of) $B$ if there is a one-to-one mapping from $A$ to (part of) $B$, but $A$ and $B$ do not have the same cardinality.

Using the Axiom of Choice, one can prove that for any two sets $A$ and $B$, either (i) $A$ and $B$ have the same cardinality or (ii) $A$ has cardinality less than $B$ or (iii) $B$ has cardinality less than $A$. (This result is sometimes called Trichotomy.)

In this way, any two sets can be compared as to "size."

It turns out that not all infinite sets have the same cardinality. The famous early result is due to Cantor. Let $\mathbb{N}$ be the set of positive integers, and let $\mathbb{R}$ be the set of reals. Then $\mathbb{N}$ has cardinality less than $\mathbb{R}$. So, in the sense of cardinality, two infinite sets can have different sizes.

In general, the collection of all subsets of a set $S$ can be proved to have cardinality greater than the cardinality of $S$. In particular, this means that the collection of all subsets of the reals has cardinality greater than the set of reals.

In the sense of cardinality, there is a very rich family of different-sized "infinities."

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Cantor proved that there are "infinities" "bigger" than others. For instance, $\mathbb{R}$ is strictly bigger than $\mathbb{N}$. What is meant by this can be stated the following way : There is not enough natural numbers to number every real number. In other words, if you have associated a real number to each natural number, then there will be real numbers left without an associated natural number.

The proof is called the Cantor Diagonal argument : http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument.

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