How does mathematical induction follow from the Peano approach to natural numbers?

In the Peano axioms, mathematical induction is not merely a consequence of the other axioms; it is explicitly defined as the fifth axiom of the system. This axiom states that if a set contains zero and the successor of every number in that set is also included, then the set must contain all natural numbers.1

Because this principle is foundational to the Peano approach, it allows mathematicians to establish properties for an infinite number of cases without requiring an infinite number of individual proofs. By demonstrating that a property holds for zero (the base case) and that the property is "hereditary"—meaning if it holds for a number, it must also hold for its successor—one can conclude that the property applies to all natural numbers.12

While some mathematicians, such as Henri Poincaré, argued that this principle is a synthetic a priori truth irreducible to pure logic, others like Gottlob Frege and Bertrand Russell sought to prove it is analytic by reducing it to logical principles.2