That one is neat, but how do you define addition and multiplication on these structures?

You use (transfinite, if your ordinals are large enough) recursion! Just define a + 1 to be the successor of a — succ(a) — and then, assuming we’ve defined a + b, define

a + (b + 1) := (a + b) + 1 = succ(a + b)

(it’s only slightly more complicated for infinite ordinals)

You can do a similar thing for multiplication, and exponents, and so on.

Technically, you have to use induction to prove that this definition indeed works to define the operations for all ordinals.