Kaye says the following in his book about models of $\textbf{PA}$ on p. 198:
I have no clue what motivates the definition of $f_n(x, y)$. The reference made to Propositions 14.1, 14.2 don't help me at all. Can anyone explain where this definition comes from and why it makes sense?
I find a lot of the material (both theorems and their proofs) in Kaye after chapter 6 to be quite poorly motivated (or is that just me?). So general advice about where to find proper motivation, particularly about the concept of indicators, is also very welcome!