I am having difficulty interpreting the definition of addition provided in Hardy' Course of Pure Mathematics.
(i) Addition. In order to define the sum of two numbers α and β, we consider the following two classes: (i) the class (c) formed by all sums $c = a+b$, (ii) the class (C) formed by all sums $C = A+B$. Plainly $c < C$ in all cases.
α and β are the points of two sections with a and b being the lower classes and A and B being the upper classes.
In case I left out a critical detail, here is the link to an online copy. The page number is 18.
My confusion is how to interpret $c=a+b$ and $C=A+B$. What I don't understand is how you can add two sections.