Transfinite Numbers in Set Theory Since my school doesn't teach higher-level things like set theory or transfinite numbers, I have been learning on my own. There are large gaps in my knowledge of higher-level math because of this.
Where would transfinite numbers fit in this sequence?
$$\mathbb S \supseteq \mathbb O \supseteq \mathbb H \supseteq \mathbb C \supseteq  \mathbb R \supseteq \mathbb Q \supseteq \mathbb Z \supseteq \mathbb N$$
or alternatively:
Sedenions $\supseteq$ Octonions $\supseteq$ Quaternions $\supseteq$ Complex numbers $\supseteq$ Real numbers $\supseteq$ Rational numbers $\supseteq$ Integers $\supseteq$ Natural numbers.
(I know I'm not including irrational numbers, but you get the point).
 A: They would fit nowhere. In fact, the "transfinite numbers", more commonly called ordinals, extend the natural numbers, but do not contain the negative integers. So, the closes thing we can say is that
$$
\mathbb S \supseteq \mathbb O \supseteq \mathbb H \supseteq \mathbb C \supseteq  \mathbb R \supseteq \mathbb Q \supseteq \mathbb Z \supseteq \mathbb N \subseteq \mathsf{On},
$$
which doesn't help you much.
In fact, the ordinals are a totally different way of extending numbers than the extensions you've given. The extensions you've given are all about having different algebraic properties; the naturals you start with, the integers solve all equations of the form $a + x = b$, the rationals all equations of the form $a\cdot x = b$, the reals complete the rationals (which gives you calculus), the complex numbers add solutions to $x^2 = -1$, and the final ones are all interesting because they extend the complex numbers in such a way that they are "finite-dimensional complex algebras" (you'll learn eventually what this means) where multiplication acts relatively nicely. By contract, the ordinals are an extension that have to do with ways of "well-ordering" sets.
If you want something in the middle, there is the field of surreal numbers -- it extends both $\mathbb R$ and $\mathsf{On}$ at the same time, and it is a very interesting object. It is in some sense the "largest field" that is still linearly ordered -- that is, either $a < b$ or $a = b$ or $a > b$. (Note that that already doesn't make sense for the complex numbers.)
A: Both the ordinal numbers and the cardinal numbers extend the natural numbers, but in a different direction than the integers, and therefore cannot be inserted into this sequence.
The ordinal and cardinal numbers add numbers that are larger than any natural numbers. In contrast, the integers add numbers smaller than all natural numbers. The rational numbers and real numbers add something in between the integers. And the complex numbers, octonions and sedenions add numbers in directions orthogonal to the real numbers.
