models of the Peano axioms (including the second-order induction axiom) are isomorphic. In particular, given two models and of the Peano axioms, there is a unique homomorphism satisfying and it is a bijection. This means that the second-order Peano axioms are categorical. (This is not the case with any first-order reformulation of the Peano axioms, below.) Set-theoretic models The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as ZF. The standard construction of the naturals, due to John von Neumann, starts from a definition of 0 as the empty set, ∅, and an operator s on sets defined as: The set of natural numbers N is defined as the intersection of all sets closed under s that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it: and so on. The set N together with 0 and the successor function satisfies the Peano axioms. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Another such system consists of general set theory (extensionality, existence of the empty set, and the axiom of adjunction), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets. Interpretation in category theory The Peano axioms can also be understood using category theory. Let C be a category with terminal object 1C, and define the category of pointed unary systems, US1(C) as follows: The objects of US1(C) are triples where X is an object of C, and and are C-morphisms. A morphism φ : (X, 0X, SX) → (Y, 0Y, SY) is a C-morphism with and . Then C is said to satisfy the Dedekind–Peano axioms if US1(C) has an initial object; this initial object is known as a natural number object in C. If is this initial object, and is any other object, then the unique map is such that This is precisely the recursive definition of 0X and SX. Consistency When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number". Henri Poincaré was more cautious, saying they only defined natural numbers if