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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?

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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
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

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.

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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.

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Cantor's diagonal method is about more than just comparing the set of all reals and the set of all rationals. – Doug Spoonwood Dec 14 '12 at 15:55

(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.

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