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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.

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up vote 2 down vote accepted

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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