Can all axioms of mathematical theories be expressed with predicate logic?

Question

The book Roads to Infinity: The Mathematics of Truth and Proof stated that, "All the standard mathematical theories have axioms that can be expressed in predicate logic." Predicate logic generally refers to any formal system like first-order, second-order logic, many-sorted logic, or infinitary logic, but it sometimes refers to just first-order logic (Wikipedia). The book never specified which it was referring to. I understand that not all axioms can be expressed in first-order logic, but what about in second-order logic, many-sorted logic or infinitary logic?