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I have heard that there are infinities of various sizes. I was wondering what that actually means-how do we compare their cardinalities? I have just started real analysis and I am slowly coming to terms with notions of countability,Cantor's diagonalization method and limit points.Can anyone please explain, in simple words,what that actually means?

Richard Nash
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    Have you looked around the site? I am fairly certain this question was answered before. – Asaf Karagila Dec 14 '12 at 13:17
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    You might want to look at this: http://math.stackexchange.com/questions/5378/types-of-infinity?rq=1 , and http://math.stackexchange.com/questions/1/different-kinds-of-infinities – Amzoti Dec 14 '12 at 13:23
  • And also http://math.stackexchange.com/questions/182171/are-all-infinities-equal – Asaf Karagila Dec 14 '12 at 13:29
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    If you had two finite piles of beans, and you didn't know how to count, how would you tell which pile had more beans? – Thomas Andrews Dec 14 '12 at 13:36
  • @ThomasAndrews You'd find two pots of water of equivalent size. You put the bean piles in separate pots. Whichever displaces more water either has more beans or almost surely has more food value, and that way you've started soaking the beans so you can cook them later. Alright, so what's a better example where one-to-one pairing might work out as a little more relevant? – Doug Spoonwood Dec 14 '12 at 15:54
  • @DougSpoonwood That doesn't work, because you don't know each bean is the same size. You don't want the weight or volume, you want to know they have the same number. (Might be easier to imagine if they were coins of the same denomination, but possibly different weights/sizes due to when they were minted.) Hint: When I said "you can't count," assume you can count to $1$ - that is, you know how to take one bean off a pile. – Thomas Andrews Dec 14 '12 at 15:56
  • @ThomasAndrews I edited my comment and added "or almost surely has more food value." You're right that doesn't take into account weight or volume and that makes using pots irrelevant for determining the number of beans. But, it doesn't make such a method irrelevant for determining how much value the piles of beans have in terms of nutrition... why else would we want to know how many beans we have in a pile? – Doug Spoonwood Dec 14 '12 at 16:04
  • @DougSpoonwood You are being obtuse, since in practicality, you can also count them, so what is the value of not knowing how to count? The point is to notice that we can determine which pile has more beans without having an ability to "count," so that the relative size of finite sets is independent of our ability to count. Then we later apply that same idea, with variations, to infinite sets. – Thomas Andrews Dec 14 '12 at 16:09
  • http://www.youtube.com/watch?v=UPA3bwVVzGI – Quinn Culver Dec 14 '12 at 16:42
  • @ThomasAndrews For finite sets if we pair off each element of set A with an element of set B which has no more than one element of A which it got paired with, and there exists at least one more element of set B which did not get paired off with an element of set A, then set B has more members. This does NOT work for the idea of cardinality in Cantorian set theory. If you pair the natural number 1 with the rational number 1, 2 with 2, and so on, the above method valid for finite sets would indicate the rationals as having a larger size than the naturals. That is NOT Cantorian cardinality. – Doug Spoonwood Dec 14 '12 at 17:06
  • @DougSpoonwood As I said, "with variations." The point is to start with the idea of comparing sizes without counting in the finite case, before jumping into the complexity of Cantor. (Without choice, two arbitrary sets aren't even necessarily comparable. Indeed, the "pairing up one at a time" is, in the infinite case, also assuming we can well-order the sets of beans, on some level, and we are really comparing the ordinals if we were to really extend my idea to infinite sets.) – Thomas Andrews Dec 14 '12 at 17:10

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Here's a relatively simple example of why this sort of distinction in "sizes" of infinite sets is important.

Let $f(x)$ be an increasing function on $[0,1]$. What can we say about the set of points on which $f(x)$ is discontinuous? That is, the set $\{a\in [0,1]:\lim_{x\to a-} f(x)\neq \lim_{x\to a+} f(x)\}$?

It turns out that, given any sequence of real numbers, $x_1,x_2,...,x_n,...\in[0,1]$, you can pretty easily construct an increasing function that is discontinous at all $x_i$.

So you might answer, "we can make one with infinitely many discontinuities!"

But it turns out, you cannot create an increasing function that is discontinuous at all points of $[0,1]$. This turns out to be precisely because the cardinality of $[0,1]$ is "bigger" than the cardinality of $\mathbb N=\{1,2,3,...\}$.

It is sometimes useful initially not to think about "different sizes of infinity." It can also be thought of in terms of "complexity." That is, the set of real numbers in $[0,1]$ is infinite in a more complicated way than the set of natural numbers, $\mathbb N$. That the definition of this "complexity" matches our notion of "size" in finite sets will lead you, with some experience, to thinking of this "complexity" as a generalization of the "size" of finite sets.

Thomas Andrews
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What does it mean that two sets $A$ and $B$ have the same size? Mathematicians agree that this means that it is possible to find a bijection $$ A\longleftrightarrow B. $$ Thus, the existence of infinities of various sizes means that it is possible to find $A$ and $B$ both infinite which do not admit a bijection as above. The goal of Cantor's diagonal method is to show that this is the situation when one takes $A=\Bbb Q$ the set of rational numbers, and $B=\Bbb R$ the set of all real numbers.

Andrea Mori
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(I'm probably assuming the axiom of choice)

Cardinality is motivated by comparing the sets element by element. If there is a bijective function $A \to B$, then we should have $|A| = |B|$. Furthermore, if there is an injective function $A \to B$, we want $|A| \leq |B|$, and if there is a surjective function $A \to B$, we want $|A| \geq |B|$.

Then, we learn how to do arithmetic with cardinal numbers, which makes it that much easier to compute and compare cardinal numbers.

However, there are many different notions of size. You might look at the properties of cardinality and think they are weird. This often means that cardinality is measuring something different than what you are thinking of; people often have something geometric in mind (e.g. a measure, or maybe the asymptotic density of a set of integers).

In some other contexts, cardinality might be thought of as more of a measure of complexity than size; e.g. the real numbers are too complicated for there to be a bijection between them and the natural numbers.

But analogies are only useful if the ideas you are drawing from the analogy are the similarities. Like most concepts, the most reliable way to learn it is through experience: use cardinality and see what you can do with it and see what you can't do with it. Every theorem is telling you something you should understand about cardinality. Every proof is an example of how to use cardinality to derive interesting results.