Okay, I have no idea if this is the proper forum to post this question, but here goes:

In most elementary mathematics texts they start off with a description of the various sets of numbers, starting with the counting numbers and working their way up to the complex numbers. Each successive set includes the sets before it, like this:

Counting numbers $\displaystyle \Rightarrow$ Natural numbers $\displaystyle \Rightarrow$ Integers $\displaystyle \Rightarrow$ Rational numbers $\displaystyle \Rightarrow$ Real numbers $\displaystyle \Rightarrow$ Complex numbers

I remember my first question in Algebra 1 when this was taught was, "What comes after that?" Of course everyone laughed and the teacher didn't know, but it seems an obvious question. How far does this sequence extend? Ie, what is the broadest possible definition of "number"?

On Wikipedia there are some tantalizing descriptions of "quaternions", "octonions", etc., themselves subsets of the "hypercomplex" numbers, and from what I can gather from the article, everything is a subset of the "surreal" numbers. So, because I'm stupid and because technical articles tend to assume prior knowledge, could someone just tell me if this is right? If you were to fill in the chart of the major number sets (each of which subsumes the previous one) from complex numbers to surreal numbers, what would it look like?

Also, does anyone have anything philosophical to say about this and what it means? Unfortunately I have only an elementary knowledge of mathematics but I find the topic fascinating.