categoricity

For the arithmeticity of the axioms of PA—i. e. the claim that they can be seen as true on the basis of our basic grasp of the structure of the natural numbers—is motivated by Isaacson by appealing to the categoricity of PA 2, the second-order theory which provides us with a categorical characterization of the natural numbers as the smallest structure closed under a one-to-one successor operation and containing an element which is not the successor of any element.