In Terence Tao's Analysis I, he admits to leaving out some details when writing about induction in his presentation of Peano Arithmetic (note that he uses $n++$ to denote the successor of $n$):
Axiom 2.5 (Principle of mathematical induction). Let $P(n)$ be any property pertaining to a natural number $n$. Suppose that $P(0)$ is true, and suppose that whenever $P(n)$ is true, $P(n++)$ is also true. Then $P(n)$ is true for every natural number $n$.
Remark 2.1.10. We are a little vague on what “property” means at this point, but some possible examples of $P(n)$ might be “$n$ is even”; “$n$ is equal to $3$”; “$n$ solves the equation $(n + 1)^2 = n^2 + 2n + 1$”; and so forth. Of course we haven’t defined many of these concepts yet, but when we do, Axiom 2.5 will apply to these properties. (A logical remark: Because this axiom refers not just to variables, but also properties, it is of a different nature than the other four axioms; indeed, Axiom 2.5 should technically be called an axiom schema rather than an axiom - it is a template for producing an (infinite) number of axioms, rather than being a single axiom in its own right. To discuss this distinction further is far beyond the scope of this text, though, and falls in the realm of logic.)
Trying to better understand this axiom (schema) and what is meant by "property", I navigated to the Wikipedia page for the Peano axioms, where it is written,
The axiom of induction is in second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers), but it can be transformed into a first-order axiom schema of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom.
My understanding is that in an interpretation of Peano Arithmetic, each "property", i.e., (single-place) predicate is interpreted by a Boolean-valued function on the domain of discourse.
But then how can one go about discussing all possible predicates without referencing a specific domain of discourse? In order to know the possible predicates, don't we need to first fix a domain of discourse? But this is getting into interpretations, whereas I was trying to state axioms without reference to an interpretation. Or should we just restrict the predicate symbols in our language to some fixed set of symbols, e.g., $P_0, P_1, P_2, \dotsc$, and then apply the axiom schema of mathematical induction to only those symbols, thereby avoiding reference to specific interpretations?