Not a professional mathematician, but you have the benefit of set operations mapping to functions you're familiar with.
For example, set union becomes the max function.
For example, set union becomes the max function.
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.