As Andy have said in class, there are two kinds of infinity. That is, the cardinality of $\mathbb{R}$ is greater than the cardinality of $\mathbb{N}$. In other words, the set of real number is uncountable. Georg Cantor proved this in 1891 by using a so-called diagonal method. The idea is that suppose that the real number is countable. Then we can list them one by one, say $s_1, s_2, s_3, ...$. Now we will use this list to construct another real number, say $s_0$ that cannot be in this list, so the list is in fact incomplete. The way to construct the number is that the first digit after the decimal point of $s_0$ is not the same as the first digit after the decimal point of $s_1$, and the second digit after the decimal point of $s_0$ is not the same as the second digit after the decimal point of $s_2$, and so on. Note that $s_0$ can't be anywhere in the list because the i^{th} digit of $s_0$ differs from the i^{th} digit of $s_i$ for any i, so the list is indeed incomplete, which means the set of real number is uncountable.

There are actually a lot more than two. In particular for any nonempty set X, the power set P(X), which is the set of all subsets of X, has a cardinality which is strictly greater than the cardinality of X. So you actually lots of different kinds of infinity.

I should add that the proof of this is basically just Cantor's Diagonal argument in another form. Kind of the same technique.

I always thought the different sizes of infinity were interesting because when it comes to computing infinite sums, I always thought that would have been cool to compute an infinite sum to a LARGER infinity. Is it possible to develop an equation that could have a different answer at a different infinity?? I think that could be something that justifies different infinities.

Does any one know if the infinite countable sets, such as the rational numbers, have the smallest cardinality of infinity? Or is there smaller? For example, the primes have a much smaller number in them than the Naturals, but they are also "countable". Does this mean they have the same cardinality as the Naturals and Rationals? How many cardinalities of infinity are there?

I am pretty sure the naturals has the smallest infinite cardinality… It is designated $\aleph_0$.

Yes, that's right. $\aleph_0$ is the smallest infinite cardinal, and so the set of primes and the rationals and all those countably infinite sets all have the same cardinality.