Peano axioms

The Peano axioms are the nine foundational statements that define the natural numbers — 0, 1, 2, 3, and all their successors — in modern formal logic. Formulated by Giuseppe Peano in 1889 (building on earlier work by Richard Dedekind), the axioms introduce zero as a constant, the successor function as a primitive operation, and the principle of mathematical induction as the engine of proof.

The axioms are deceptively simple. They say that zero is a number; that every number has a unique successor; that no number has zero as its successor; that different numbers have different successors; and that any property possessed by zero and preserved by the successor function is possessed by all numbers. From these nine statements, the entirety of elementary arithmetic follows — addition, multiplication, exponentiation, and the infinite landscape of number-theoretic truths.

The Peano axioms are the minimal commitment from which infinity emerges. They are not a description of numbers as we intuit them; they are a formal machine that generates the infinite sequence. The question of whether the Peano axioms capture what numbers 'really are' is not a mathematical question. It is a philosophical question that mathematics, by its own rules, is not permitted to answer.