Completeness is good to have, and for this reason, both the Peano and the Real numbers axioms, which by themselves are second-order logic, are usually stated within set theory, where they become first-order logic.

But then my question is: when axioms are in second-order logic, it is not true that "any logic deduction from a set of axioms is equivalent to proving the theorem for all models satisfying the axioms"?

Is there an example of a logical deduction from a set of axioms (written in second-order logic), such that the resulting theorem is not valid for at least one model satisfying the axioms?