How do the Peano axioms relate to set theory?

The Peano axioms and set theory are deeply interconnected, as both provide rigorous foundations for the natural numbers.

While the Peano axioms introduce natural numbers by postulating the properties of zero and a successor operation, set theory provides a structural framework to construct these numbers. In standard Zermelo-Fraenkel set theory, the Peano postulates can be derived by identifying natural numbers with specific sets. For example, zero is defined as the empty set, and each subsequent number is defined as the set containing its predecessors (e.g., 1 is the set containing the empty set).12

The axiom of infinity in set theory is essential for this relationship, as it guarantees the existence of an infinite set, allowing for the construction of the entire sequence of natural numbers. Consequently, set theory serves as a powerful environment where the properties of arithmetic, originally defined by Peano, are formally realized and extended into the study of ordinal numbers.134