The assumptions that gave the counting numbers a logical foundation
Assumptions for the Counting Numbers
The Peano axioms — also called the Dedekind–Peano axioms or the Peano postulates — are a set of axioms for the natural numbers. Giuseppe Peano, an Italian mathematician and a founder of mathematical logic and set theory, presented them in 1889 as a collection of axioms in his Latin-language book Arithmetices principia, nova methodo exposita, whose title translates as The principles of arithmetic presented by a new method. His list was a simplified version of an axiomatization Richard Dedekind had put forward in 1888; an earlier axiomatization of natural-number arithmetic had been given by Charles Sanders Peirce in 1881. The ground had been prepared by Hermann Grassmann, who in the 1860s showed that many facts of arithmetic could be derived from more basic facts about the successor operation and induction — at a time when the value of formalizing arithmetic was not widely recognized.
Zero, Successor, and the Shape of the Numbers
The axioms describe the arithmetical properties of the natural numbers, usually written N. They rely on just two non-logical symbols: a constant, 0, and a unary function, S, the successor function. The first axiom states that 0 is a natural number. Four further axioms concern equality; because they are logically valid in first-order logic with equality, modern treatments usually treat them as part of the underlying logic rather than as part of the Peano axioms themselves. Three axioms then state basic first-order facts about succession: the successor of a natural number is a natural number, 0 is not the successor of any natural number, and no two distinct natural numbers share the same successor.
The ninth and final axiom is a second-order statement of the principle of mathematical induction. In its usual form it says that if a collection of natural numbers contains 0 and contains the successor of every number it contains, then it contains every natural number. This is the axiom that guarantees the numbers obtained by repeatedly applying S to 0 — 1 as S(0), 2 as S(S(0)), and so on — exhaust the natural numbers. Without it, the other axioms do not rule out additional elements lying beyond the reach of succession.
Peano’s original formulation used 1 rather than 0 as the first natural number; the version of the axioms in his later Formulario mathematico includes zero.
Building Arithmetic from the Axioms
Once the axioms are accepted, the familiar operations can be constructed. With the second-order induction axiom in place, addition, multiplication, and a total (linear) ordering on N can be defined directly from the axioms. With only first-order induction this is not possible, and addition and multiplication are often added as axioms; either way, the functions and relations are built in set theory or second-order logic and shown to be unique using the Peano axioms.
Addition is introduced recursively: a + 0 = a, and a + S(b) = S(a + b). From these two clauses, induction yields the commutativity of addition. The resulting structure is a commutative monoid with identity 0; it is also cancellative, so it embeds in a group, and the smallest such group is the integers. Multiplication is defined recursively in the same spirit, with a · 0 = 0 and a · S(b) = a + (a · b). Here S(0), that is, 1, is the multiplicative identity, and induction establishes that multiplication is commutative and distributes over addition. Together, addition, multiplication, 0 and 1 give the natural numbers the structure of a commutative semiring.
A total order can be defined by setting a ≤ b exactly when some natural number c satisfies a + c = b. This order is preserved by both addition and multiplication, making the natural numbers an ordered semiring; since no natural number lies strictly between 0 and 1, it is a discrete ordered semiring. A stronger induction principle, known as strong induction, follows from the standard axiom and is often more convenient when reasoning about ≤. With it one can prove that the naturals are well-ordered: every nonempty set of natural numbers has a least element.
A Symbolic Language and Two Versions of the System
When Peano wrote down his axioms, the language of mathematical logic was still in its infancy. The logical notation he created to present them never became popular, yet it gave rise to the modern symbol for set membership, ∈, which descends from Peano’s ε. Peano was careful to separate mathematical from logical symbols, a distinction that was not yet common in mathematics; it had first been drawn in Gottlob Frege’s Begriffsschrift of 1879. Peano, unaware of Frege’s work, independently rebuilt a logical apparatus on foundations laid by George Boole and Ernst Schröder.
The original induction axiom is second-order. It is now common to replace it with a weaker first-order induction scheme, and the resulting system — formed by explicitly adding symbols for addition and multiplication and substituting the first-order schema for the second-order axiom — is what is often meant specifically by Peano arithmetic. The axiomatization of arithmetic as a whole is also commonly called Peano arithmetic, though the term can thus refer either to the full system or to this restricted first-order one. The two formulations differ in important ways.
One Structure, up to Isomorphism
A model of the Peano axioms is a triple (N, 0, S) in which N is a necessarily infinite set, 0 is an element of N, and S is a function from N to N satisfying the axioms. Dedekind showed in his 1888 book on the nature and meaning of numbers that any two models of the axioms, second-order induction included, are isomorphic: there is a unique structure-preserving map from one to the other, and it is a bijection. The second-order Peano axioms are therefore categorical — they pin down a single structure up to isomorphism. No first-order reformulation has this property.
The axioms can also be obtained from set theory. In the standard construction due to John von Neumann, 0 is the empty set and the successor of a set a is a ∪ {a}; each natural number then coincides with the set of natural numbers smaller than it. Category theory offers a further perspective: a category is said to satisfy the Dedekind–Peano axioms when its category of pointed unary systems has an initial object, an object known as a natural number object, whose defining property is precisely the recursive definition of zero and successor.
The Question of Consistency
Early readers treated the axioms as an implicit definition of what a natural number is; Bertrand Russell and others agreed with this reading. Henri Poincaré was more cautious, observing that the axioms define the natural numbers only if they are consistent, since a derivation of a contradiction such as 0 = 1 would leave them defining nothing. In 1900 David Hilbert made the problem of proving the consistency of arithmetic by finitistic methods the second of his twenty-three problems. Kurt Gödel’s second incompleteness theorem of 1931 then showed that such a consistency proof cannot be carried out within Peano arithmetic itself, assuming Peano arithmetic is consistent.
Gödel himself pointed out that consistency might still be established by finitistic methods that are not formalizable in Peano arithmetic, and in 1958 he published a method for proving the consistency of arithmetic using type theory. In 1936 Gerhard Gentzen gave a consistency proof for Peano’s axioms using transfinite induction up to an ordinal called ε0. Whether Gentzen’s argument meets Hilbert’s requirements remains unclear, since there is no generally accepted definition of a finitistic proof and Hilbert never supplied a precise one. The great majority of contemporary mathematicians believe the axioms to be consistent, relying either on intuition or on accepting a proof such as Gentzen’s.
A Framework Used Nearly Unchanged
The Peano axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. That durability — a small set of assumptions surviving as the standard foundation for the natural numbers — is the measure of the system Peano set out in his Latin treatise.
Giuseppe Peano was also a glottologist. He was the author of over 200 books and papers, and he contributed much notation to mathematical logic and set theory, including notations for set operations. He spent most of his career teaching mathematics at the University of Turin. He created an international auxiliary language, Latino sine flexione (“Latin without inflections”), a simplified version of Classical Latin; most of his books and papers are in Latino sine flexione, while others are in Italian.
Peano was born and raised on a farm at Spinetta, a hamlet now belonging to Cuneo, Piedmont, Italy. He attended the Liceo classico Cavour in Turin and enrolled at the University of Turin in 1876, graduating in 1880 with high honors. The university employed him to assist first Enrico D’Ovidio and then Angelo Genocchi, the Chair of calculus. Because of Genocchi’s poor health, Peano took over the teaching of the calculus course within two years. His first major work, a textbook on calculus, was published in 1884 and credited to Genocchi. A few years later, Peano published his first book dealing with mathematical logic; there the modern symbols for the union and intersection of sets appeared for the first time.
In 1886, he began teaching concurrently at the Royal Military Academy. In 1887, Peano married Carola Crosio, the daughter of the Turin-based painter Luigi Crosio, known for painting the Refugium Peccatorum Madonna. He was promoted to Professor First Class in 1889. The next year, the University of Turin granted him his full professorship. Peano’s space-filling curve appeared in 1890 as a counterexample; he used it to show that a continuous curve cannot always be enclosed in an arbitrarily small region, and it was an early example of what came to be known as a fractal. In 1890 he founded the journal Rivista di Matematica, which published its first issue in January 1891. In 1891 he started the Formulario Project, intended as an “Encyclopedia of Mathematics” containing all known formulae and theorems of mathematical science using a standard notation invented by Peano. In 1897, the first International Congress of Mathematicians was held in Zürich; Peano was a key participant, presenting a paper on mathematical logic, and he also started to become increasingly occupied with Formulario to the detriment of his other work. In 1898 he presented a note to the Academy about binary numeration and its ability to be used to represent the sounds of languages.
Proof Details for Addition, Multiplication, and Order
In the derivation of commutativity of addition, one first proves 0 + b = b and S(a) + b = S(a + b), each by induction on b; using both results, one then proves a + b = b + a by induction on b. The right multiplicative identity follows directly: a · S(0) = a + (a · 0) = a + 0 = a. Showing that S(0) is also the left multiplicative identity requires the induction axiom: S(0) · 0 = 0; and if S(0) · a = a, then S(0) · S(a) = S(0) + (S(0) · a) = S(0) + a = a + S(0) = S(a + 0) = S(a), using commutativity of addition. Therefore, by the induction axiom, S(0) is the multiplicative left identity of all natural numbers.
Strong induction can be stated with the order relation: for any predicate φ, if φ(0) is true and if, for every n ∈ N, φ(k) being true for every k ∈ N with k ≤ n implies φ(S(n)) is true, then φ(n) is true for every n ∈ N. From this, well-ordering can be proved as follows: let a nonempty X ⊆ N be given and assume X has no least element. Because 0 is the least element of N, 0 ∉ X. For any n ∈ N, suppose every k ≤ n is not in X; then S(n) ∉ X, for otherwise it would be the least element of X. Thus, by strong induction, every n ∈ N is not in X, so X ∩ N = ∅, contradicting that X is a nonempty subset of N. Hence X has a least element.
Equiconsistency with Weak Set Theories
Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Another consists of general set theory—extensionality, existence of the empty set, and the axiom of adjunction—augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets.
Rejections and Self-Verifying Variants
A small number of philosophers and mathematicians, some of whom also advocate ultrafinitism, reject Peano’s axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers. In particular, addition (including the successor function) and multiplication are assumed to be total. Curiously, there are self-verifying theories that are similar to PA but have subtraction and division instead of addition and multiplication, axiomatized in such a way to avoid proving sentences that correspond to the totality of addition and multiplication, but which are still able to prove all true Π1 theorems of PA, and yet can be extended to a consistent theory that proves its own consistency (stated as the non-existence of a Hilbert-style proof of “0=1”).
Finitism and Gentzen’s Proof
Although it is widely claimed that Gödel’s theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gentzen explained: “The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles”. Gentzen’s proof is arguably finitistic, since the transfinite ordinal ε0 can be encoded in terms of finite objects—for example, as a Turing machine describing a suitable order on the integers, or more abstractly as consisting of the finite trees, suitably linearly ordered.
