Peanoaxiomerne

Peanoaxiomerne, or the Peano axioms, are a foundational set of principles intended to formalize the natural numbers and basic arithmetic. They were introduced by the Italian mathematician Giuseppe Peano in the late 19th century as part of an effort to ground arithmetic in a precise logical system. The axioms are usually presented in a first‑order framework and underpin Peano arithmetic (PA).

In their common form, the axioms describe a starting point and a successor operation. They typically include: 0 is a natural number; every natural number has a unique successor; 0 is not the successor of any natural number; distinct numbers have distinct successors (the successor function is injective); and an induction axiom schema: for any property P, if P holds for 0 and P(x) implies P(S(x)) for all x, then P holds for every natural number. Most presentations use 0 as the starting element, though some variants start with 1. Addition and multiplication are defined from the successor function or added as primitive notions within PA.

Peano axioms serve as the standard formal basis for elementary number theory and the formalization of arithmetic. They emphasize a constructive, recursive view of numbers and support proofs about basic properties of addition and multiplication. In modern logic, PA is known to be sound relative to the standard model of natural numbers, but it is incomplete and has nonstandard models due to the nature of first‑order logic. The framework has influenced foundational studies in mathematics and logic for over a century.