Properties are sets of numbers, so without getting into technicalities, you need second-order logic to talk about the smallest set such that whatever (since you need to quantify over all candidate sets).
Similarly, to say that you can get x by iterating the successor function on zero requires second-order logic. First-order logic isn’t even sufficient to define addition without adding axioms for what addition does.
Properties are sets of numbers, so without getting into technicalities, you need second-order logic to talk about the smallest set such that whatever (since you need to quantify over all candidate sets).
Similarly, to say that you can get x by iterating the successor function on zero requires second-order logic. First-order logic isn’t even sufficient to define addition without adding axioms for what addition does.