Peano Axioms

The Peano axioms are a set of foundational axioms that formally define the natural numbers and their basic arithmetic properties, providing a rigorous logical basis from which the rest of elementary number theory can be derived. Italian mathematician Giuseppe Peano first published the axioms in a formal, symbolic notation in an 1889 book, building on earlier informal work by mathematician Richard Dedekind. The axioms establish that zero (or one, in some formulations) is a natural number, that every natural number has a unique successor which is also a natural number, that zero is not the successor of any natural number, that distinct natural numbers have distinct successors, and that mathematical induction holds, meaning any property true of zero and preserved by the successor operation is true of every natural number. The Peano axioms remain the standard formal starting point in mathematical logic and set theory for constructing the natural numbers and proving their basic properties rigorously, rather than simply assuming them as an intuitive, pre-formal notion.