How do I refer to a certain element in a set? I have an idea about how to do this. I've seen this before so this is what I think. Is this how you refer to a single element in a set?
$$a = \{5, 7, 3, 4\};a_2 = 7$$
Is this correct?

You can just refer to it as 7. To say 7 is an element of the set, we write $7 \in \{5,7,3,4 \}$. Usually the order you write elements of sets in doesn't matter. 
You can have very big (uncountable) sets where it is not easy to assign a number to each element of the set, for example the set of real numbers (i.e. decimal numbers) $\mathbb{R}$. So there is no (immediate) sensible idea of a "2nd element" of this set, but there is for your example.
