Axiom schemas, such as the PA induction schema, differ from second-order axioms in that they only hold for "definable" predicates, rather for "all" predicates. As a result, you can have non-standard models of first-order PA for which induction holds for all definable predicates, but does not hold true in general.
This leads to questions about which predicates "exist." In general, for every possible infinite combination of natural numbers (or more generally, objects in the domain), is there considered to "exist" a unary predicate that is "true" for only those numbers, even if the predicate is undefinable? And likewise with n-ary predicates?
This would seem to be a semantic thing. Is there an "axiom of first-order logic" that determines which of these predicates we consider to "exist?" (Or would this perhaps be more pertinent to second-order logic, being related to things like Henkin semantics?)