$ \newcommand{\N}{\mathbb{N}} $ For example, the well-ordering principle (WOP) states that every nonempty subset of $\N$ has a least element. In symbols: $$ \forall A \subseteq \N: [A \neq \varnothing \Rightarrow \exists x \in A:\forall y \in A: x \le y] $$ It quantifies over the subsets of $\N$. I've seen that the statement is second-order (SO) because the set of all subsets of $\N$ cannot be expressed in the first-order (FO) logic.
On the other hand, the axiom of power set in ZFC based on FOL, states that every set has a power set. The quantification over the subsets is just the quantification over the element of the power set, i.e., $\forall A \in \mathcal{P}(\N):\phi(A)$. Statements involving the quantification over the elements of sets, which are also sets, is FO.
On this (possibly incorrect) base knowledge, why does WOP has to be SO? I guess the reason is that the construction of the set of all subsets of $\N$ infeasible in FO in the first place. As a result, $\forall A \subseteq \N:\phi(A)$ is SO.
If this observation is correct, given that the set $B$ of subsets of some set can be constructed in FOL, is the sentence $\forall A \in B: \phi(A)$ in FOL?
What introduction-level textbooks do you recommend to clarify the notions required to answer the question? The only logic book I've read so far is Barker-Plummer's Language, Proof and Logic (2011), but it does not describe much about SOL.
