In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of consistency and completeness of number theory.
The need for formalism in arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction.^{[1]} In 1888, Richard Dedekind proposed a collection of axioms about the numbers,^{[2]} and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).^{[3]}
The Peano axioms contain three types of statements. The first four statements are general statements about equality; in modern treatments these are often considered axioms of pure logic^{[citation needed]}. The next four axioms are firstorder statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker firstorder system called Peano arithmetic is obtained by replacing this secondorder induction axiom with a firstorder axiom schema.
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When Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership (symbol: ∈, from Peano's ε) and implication (symbol: ⊃, from Peano's reversed 'C'). Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the Begriffsschrift by Gottlob Frege, published in 1879.^{[4]} Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of Boole and Schröder.^{[5]}
The Peano axioms define the properties of natural numbers, usually represented as a set N or The signature (a formal language's nonlogical symbols) for the axioms includes a constant symbol 0 and a unary function symbol S.
The first four axioms describe the equality relation.^{[6]}
The remaining axioms define the properties of the natural numbers. The constant 0 is assumed to be a natural number, and the naturals are assumed to be closed under a "successor" function S.
Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as axiom 5 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 5 and 6 define a unary representation of the natural numbers: the number 1 is S(0), 2 is S(S(0)) (= S(1)), and, in general, any natural number n is S^{n}(0). The next two axioms define the properties of this representation.
These two axioms together imply that the set of natural numbers is infinite, because it contains at least the infinite subset { 0, S(0), S(S(0)), … }, each element of which differs from the rest. The final axiom, sometimes called the axiom of induction, is a method of reasoning about all natural numbers.
The induction axiom is sometimes stated in the following form:
In Peano's original formulation, the induction axiom is a secondorder axiom. It is now common to replace this secondorder principle with a weaker firstorder induction scheme. There are important differences between the secondorder and firstorder formulations, as discussed in the section Models below. Without the axiom of induction, the remaining Peano axioms give a theory equivalent to Robinson arithmetic, which can be expressed without secondorder logic.
The Peano axioms can be augmented with the operations of addition and multiplication and the usual total (linear) ordering on N. The respective functions and relations are constructed in secondorder logic, and are shown to be unique using the Peano axioms.
Addition is the function + : N × N → N (written in the usual infix notation), defined recursively as:
For example,
The structure (N, +) is a commutative semigroup with identity element 0. (N, +) is also a cancellative magma, and thus embeddable in a group. The smallest group embedding N is the integers.
Given addition, multiplication is the function · : N × N → N defined recursively as:
It is easy to see that 1 is the multiplicative identity:
Moreover, multiplication distributes over addition:
Thus, (N, +, 0, ·, 1) is a commutative semiring.
The usual total order relation ≤ : N × N can be defined as follows:
This relation is stable under addition and multiplication: for , if a ≤ b, then:
Thus, the structure (N, +, ·, 1, 0, ≤) is an ordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring. The axiom of induction is sometimes stated in the following strong form, making use of the ≤ order:
This form of the induction axiom is a simple consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are wellordered—every nonempty subset of N has a least element—one can reason as follows. Let a nonempty X ⊆ N be given and assume X has no least element.
Thus, by the strong induction principle, for every n ∈ N, n ∉ X. Thus, X ∩ N = ∅, which contradicts X being a nonempty subset of N.Thus X has a least element.
A model of the Peano axioms is a triple (N, 0, S), where N an infinite set, 0 ∈ N and S : N → N satisfies the axioms above. Dedekind proved in his 1888 book, What are numbers and what should they be (German: Was sind und was sollen die Zahlen) that any two models of the Peano axioms (including the secondorder induction axiom) are isomorphic. In particular, given two models (N_{A}, 0_{A}, S_{A}) and (N_{B}, 0_{B}, S_{B}) of the Peano axioms, the homomorphism f : N_{A} → N_{B} defined as
is a bijection. The secondorder Peano axioms are thus categorical; this is not the case with any firstorder reformulation of the Peano axioms, however.
Firstorder theories are often better than second order theories for model or proof theoretic analysis. All but the ninth axiom (the induction axiom) are statements in firstorder logic. The arithmetical operations of addition and multiplication and the order relation can also be defined using firstorder axioms. The secondorder axiom of induction can be transformed into a weaker firstorder induction schema; the first eight of Peano's axioms together with the firstorder induction schema form a firstorder axiomatization of arithmetic called Peano arithmetic (PA).
The induction schema consists of a countably infinite set of axioms. For each formula φ(x,y_{1},...,y_{k}) in the language of Peano arithmetic, the firstorder induction axiom for φ is the sentence
where is an abbreviation for y_{1},...,y_{k}. The firstorder induction schema includes every instance of the firstorder induction axiom, that is, it includes the induction axiom for every formula φ.
This schema avoids quantification over sets of natural numbers, which is impossible in firstorder logic. For instance, it is not possible in firstorder logic to say that any set of natural numbers containing 0 and closed under successor is the entire set of natural numbers. What can be expressed is that any definable set of natural numbers has this property. Because it is not possible to quantify over definable subsets explicitly with a single axiom, the induction schema includes one instance of the induction axiom for every definition of a subset of the naturals.
There are many different, but equivalent, axiomatizations of Peano arithmetic. While some axiomatizations, such as the one just described, use a signature that only has symbols for 0 and the successor operation, other axiomatizations use the language of ordered semiring, including addition and multiplication function symbols and an order relation symbol. One such axiomatization begins with the following axioms that describe a discrete ordered semiring.^{[7]}
The theory defined by these axioms is known as PA^{–}; PA is obtained by adding the firstorder induction schema.^{[8]} An important property of PA^{–} is that any structure M satisfying this theory has an initial segment (ordered by ≤) isomorphic to N. Elements of M \ N are known as nonstandard elements.
Although the usual natural numbers satisfy the axioms of PA, there are other nonstandard models as well; the compactness theorem implies that the existence of nonstandard elements cannot be excluded in firstorder logic. The upward Löwenheim–Skolem theorem shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (secondorder) Peano axioms, which have only one model, up to isomorphism. This illustrates one way the firstorder system PA is weaker than the secondorder Peano axioms.
When interpreted as a proof within a firstorder set theory, such as ZFC, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any firstorder formalization of set theory.
It is natural to ask whether a countable nonstandard model can be explicitly constructed. It is possible to explicitly describe the order type of any countable nonstandard model: it is always ω + (ω* + ω)η, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers. However, a theorem by Stanley Tennenbaum, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable.^{[9]} This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA.
The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as the ZF.^{[10]} The standard construction of the naturals, due to John von Neumann, starts from a definition of 0 as the empty set, ∅, and an operator s on sets defined as:
The set of natural numbers N is defined as the intersection of all sets closed under s that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it:
and so on. The set N together with 0 and the successor function s : N → N satisfies the Peano axioms.
Peano arithmetic is equiconsistent with several weak systems of set theory.^{[11]} One such system is ZFC with the axiom of infinity replaced by its negation. Another such system 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.
The Peano axioms can also be understood using category theory. Let C be a category with initial object 1_{C}, and define the category of pointed unary systems, US_{1}(C) as follows:
Then C is said to satisfy the Dedekind–Peano axioms if US_{1}(C) has an initial object; this initial object is known as a natural number object in C. If (N, 0, S) is this initial object, and (X, 0_{X}, S_{X}) is any other object, then the unique map u : (N, 0, S) → (X, 0_{X}, S_{X}) is such that
This is precisely the recursive definition of 0_{X} and S_{X}.
When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number". Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don't define anything. In 1900, David Hilbert posed the problem of proving their consistency using only finitistic methods as the second of his twentythree problems.^{[12]} In 1931, Kurt Gödel proved his second incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself.^{[13]}
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. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958 Gödel published a method for proving the consistency of arithmetic using type theory.^{[14]} In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε_{0}.^{[15]} 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 ε_{0} can be encoded in terms of finite objects (for example, as a Turing machine describing a suitable order on the integers). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.
The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen's proof. The small number of mathematicians who advocate ultrafinitism reject Peano's axioms because the axioms require an infinite set of natural numbers.
This article incorporates material from PA on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
