The Peano axioms, also known as the Peano–Dedekind axioms, are a set of axioms for the natural numbers presented by the Italian mathematician Giuseppe Peano in 1889. They provide a formal foundation for arithmetic by specifying the properties of the set of natural numbers ℕ and a distinguished successor function S.
Formal statement (in first‑order language with equality, a constant 0, and a unary function S):
- Existence of zero: 0 is a natural number.
- Successor function: For every natural number n, S(n) is a natural number.
- Injectivity of the successor: For all m, n ∈ ℕ, if S(m) = S(n) then m = n.
- Zero is not a successor: For all n ∈ ℕ, S(n) ≠ 0.
- Induction axiom: For any property P(x) definable in the language, if P(0) holds and ∀n (P(n) → P(S(n))) holds, then ∀n P(n) holds.
These axioms can be expressed in second‑order logic (where the induction axiom quantifies over all properties) or in first‑order logic with an infinite schema of induction axioms, one for each formula defining a property.
Historical context
- Giuseppe Peano introduced the axioms in his 1889 work Arithmetices Principia, Nova Methodo Exposita.
- The system built on earlier work by Dedekind, who formalized the concept of a simply infinite chain, and by Frege, whose logical notation influenced Peano’s formulation.
- The notation introduced by Peano (e.g., “∈” for set membership, “⊂” for subset, and the use of “ℕ” for the natural numbers) became standard in mathematical logic.
Significance
- The Peano axioms are foundational for the formal development of number theory, recursion theory, and mathematical logic.
- They provide a minimal, categorical description of ℕ in second‑order logic: any two models satisfying the axioms are isomorphic.
- In first‑order logic, the axioms are not categorical; non‑standard models exist due to the Löwenheim–Skolem theorem, a fact that motivates the study of non‑standard analysis.
- The axioms underpin the definition of primitive recursive functions and the formalization of computability.
Related concepts
- Peano arithmetic (PA): The first‑order theory consisting of the Peano axioms together with the usual axioms for addition and multiplication defined recursively.
- Second‑order Peano arithmetic: A stronger system where the induction axiom quantifies over all subsets of ℕ, yielding categoricity.
- Non‑standard models of PA: Structures that satisfy the first‑order Peano axioms but contain “infinite” natural numbers beyond the standard ℕ.
References
- Peano, G. (1889). Arithmetices Principia, Nova Methodo Exposita. Turin.
- Mendelson, E. (2015). Introduction to Mathematical Logic (6th ed.). CRC Press.
- Kaye, R. (1991). Models of Peano Arithmetic. Oxford University Press.
