 Owing to the central role that real numbers play in mathematics, the second-order theory of natural numbers, known as second-order arithmetic (Simpson 1999 [2009]), and denoted \(Z_2\), is an important foundational theory. It is stronger than (first order) Peano arithmetic but weaker than set theory. It has variables for individuals thought of as natural numbers as well as variables for sets of natural numbers thought of as real numbers. In addition there are \(+\) and \(\times\) for arithmetic operations on the individuals. As axioms \(Z_2\) has some rather obvious axioms about \(+\) and \(\times\), the Induction Axiom (\ref{ind}), and the axioms of second-order logic, including the Comprehension Principles (\ref{CA}). A surprising amount of mathematics can be derived in \(Z_2\). We refer to Simpson (1999 [2009]) for details. In a sense, \(Z_2\) is a great success story for second-order logic.