# Confusion with respect to definition of addition using sections (Hardy)

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.

http://www.gutenberg.org/files/38769/38769-pdf.pdf

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.

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In Hardy's book, we are actually adding representatives of the lower and upper classes respectively, not the sections. When we consider the new classes formed, we end up with a new section, which is how we define $\alpha+\beta$. Since the lower case letters represent members from the lower class (equivalently, the members less than the corresponding Greek letter), it makes perfect sense that $c<C$. I hope this helps!

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