# 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).

• I would have them being a chain extending from the natural numbers upwards out of the plane of the paper from the other items you've listed, because the way in which they extend the natural numbers is different than the way the sets you listed extend the natural numbers. Commented Sep 29, 2016 at 20:43

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.)

• I already understand hypercomplex numbers (I think, but I'm up for learning more), they're really fun, and what got me interesting in math. I just have large gaps in my knowledge of some parts of higher-level mathematics because I am learning on my own (sucks, wish I had more opportunities through official channels). Commented Sep 29, 2016 at 20:50
• Hypercomplex numbers are great, but understand that they don't include the ordinals either. When you are first learning about ordinals, it's probably easier to forget about the whole extension sequence you posted. They just don't have that much to do with each other! Do you know what a well-order is yet? Commented Sep 29, 2016 at 20:51
• Not at all, in fact this whole deal of ordinals and transfinite numbers is new to me. I just figured since I sorta understand set theory, if I could relate it to set theory in some way I'd understand it more. But from your answer, it seems like they are completely separate, more or less. Commented Sep 29, 2016 at 20:57
• A well-ordered set $(X, <)$ is a linearly ordered set such that every non-empty subset of $X$ has a $<$-least element. For example, the natural numbers are a well-order. The integers are not. But also two copies of the natural numbers stacked atop each other. The ordinals are basically all well-orders; each one appears in it. (The natural numbers are usually called $\omega$, and then two copies of it is called $\omega\cdot 2$. ) Commented Sep 29, 2016 at 21:00

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.