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).

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.
