matches between first and second order logic

Question

What is common matches between first order logic and the second order logic? in other word what in first order logic is in second order logic?

Answer 1

Every sentence in first-order predicate logic is also a sentence in second-order predicate logic. (Assuming the langues are compatible, i.e. you have the same predicate and function symbols, of course).

And if it's provable in first-order predicate logic, it's provable in second-order predicate logic. (Now assuming the languages are compatible, and the axioms you use in second-order predicate logic are a superset of the one in first-order predicate logic, of course).

The converse is not true, i.e. second-order predicate logic is (way!) more powerfull than first-order predicate logic. You can, for example, uniquely define the integers in second-order predicate logic, but you cannot do so in first-order predicate logic.