(before saying it's duplicate, read whole question)
I was told by someone that we can define addition and multiplication purely in terms of successor function, provided that we work in second order arithmetic. When trying to find a way in which it's defined, I found this M.SE question, and in the first answer a definition is given. I found it, however, quite unsatisfactory, because this definition uses quantification over functional symbols (which can be rewritten to use binary predicate quantification), and I expected a definition which would only require quantification over subsets of $\Bbb N$ (equivalently unary predicates). I expect something similar to answer to this question.
My question is: how to define addition in second order arithmetic using quantification over sets only? Is that even possible?
Thanks in advance.