Peano axioms

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In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic. The axioms are usually encountered in a first-order form, where the crucial second-order induction axiom is replaced by an infinite first-order induction schema, and Peano Arithmetic (PA) is by convention the name of the widely used system of first-order arithmetic given using this first-order form. However, Peano arithmetic is essentially weaker than the second-order axiom system, since there are nonstandard models of Peano arithmetic, and the only model for the Peano axioms (considered as second-order statements) is the usual system of natural numbers (up to isomorphism).

Peano first gave his axioms in a Latin text Arithmetices principia, nova methodo exposita published in 1889 (Peano 1889), where Peano gave nine axioms, four axioms specifying the behaviour of the equality relation and five rules involving the specifically arithmetic terms for zero and successor. It is the latter five rules that are usually intended when one discusses the Peano axioms. Peano took logical principles to be given.

Peano arithmetic constitutes a fundamental formalism for arithmetic, and the Peano axioms can be used to construct many of the most important number systems and structures of modern mathematics. Peano arithmetic raises a number of metamathematical and philosophical issues, primarily involving questions of consistency and completeness.

Contents

[edit] The axioms

Informally, the Peano axioms may be stated as follows:

  • 0 is a natural number.
  • Every natural number a has a successor, denoted by Sa or a'.
  • No natural number has 0 as its successor.
  • Distinct natural numbers have distinct successors: a = b if and only if Sa = Sb.
  • If a property holds for 0, and holds for the successor of every natural number for which it holds, then the property holds for all natural numbers. (This axiom of induction is also called "mathematical induction". Its point is to limit the natural numbers to just those which are required by the other axioms.)

Peano's original axioms (1889) are preceded with the definitions:

  • "The sign N means number (positive integer).
  • "The sign 1 means unity" (italics in original, van Heijenoort (1976) p. 94)

His first axiom is "1 ε N" (ibid), the ε signifying "is an element/member of". No mention is made of the sign "0" (also cf commentary by van Heijenoort p. 83, and Dedekind's Letter to Keferstein" (1890) p.100).

Letting the first natural number be 1 requires replacing 0 with 1 in the above axioms. Starting with 1 changes the recursive definitions of addition and multiplication given below, if one wishes the symbol "1" to represent what is normally considered the number "1". In the representation of unary numbers on the tape of a Turing machine, and in particular the Post-Turing machine, the place-holder "0" is frequently represented by a single tally mark " | ", often written as " 1 ", the unit by two marks " | | ", etc. (cf Davis (1974) p. 72).

More formally and following Dedekind (1888), define a Dedekind-Peano structure as an ordered triple (X, x, f), satisfying the following properties:

  • X is a set, x is an element of X, and f is a map from X to itself.
  • x is not in the range of f.
  • f is injective.
  • If A is a subset of X satisfying:
    • x is in A, and
    • If a is in A, then f(a) is in A

then A = X.

The following diagram sums up the Peano axioms:

x\mapsto f(x)\mapsto f(f(x))\mapsto f(f(f(x)))\mapsto\dotsb

where each of the iterates f(x), f(f(x)), f(f(f(x))), ... of x under f are distinct.

Without the axiom of induction, one could have closed loops of any finite length or extra copies of the integers or natural numbers added to the usual natural numbers.

[edit] Binary operations and ordering

The above axioms can be augmented by definitions of addition and multiplication over the natural numbers N, and by the usual ordering of N.

To define addition '+' recursively in terms of successor and 0, let a+0 = a and a+Sb = S(a+b) for all a, b. This turns \langle N,+\rangle into a commutative monoid with identity element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and is therefore embeddable in a group. The smallest group containing the natural numbers is the integers.

Given this definition of addition and 1 := S0, we see that: b+1 = b+S0 = S(b+0) = Sb, that is, b+1 is simply the successor of b.

Given successor, addition, and 0, define multiplication '·' by the recursive axioms a·0 = 0 and a·Sb = (a·b) + a. Hence \langle N,\cdot \rangle is also a commutative monoid with identity element 1. Moreover, addition and multiplication are compatible by virtue of the distribution law:

a·(b+c) = (a·b) + (a·c).

Define the usual total order on N, ≤, as follows. For any two natural numbers a and b, a ≤ b if and only if there exists a natural number c such that a+c = b. This order is compatible with addition and multiplication in the following sense: if a, b and c are natural numbers and a ≤ b, then a+c ≤ b+c and a·c≤b·c. An important property of N is that it also constitutes a well-ordered set; every nonempty subset of N has a least element.

[edit] Peano arithmetic

Peano Arithmetic (PA) reformulates the Peano axioms as a first order theory with two binary operations, addition and multiplication, recursively defined as in the preceding section, and denoted by infix '+' and '·', respectively. The conceptual change is that the Axiom schema of Induction is replaced by a schema permitting induction only over arithmetical formulae φ, whose non-logical symbols are just 0, S, '+', '·', and lower case letters as variables ranging over the natural numbers. The new axiom schema represents a countably infinite set of axioms.

The axioms of PA are:

  • \forall x(Sx \neq 0).
  • \forall x,y((Sx = Sy) \Rightarrow x=y).
  • (\varphi[0] \wedge \forall x(\varphi[x]\Rightarrow\varphi[Sx])) \Rightarrow \forall x(\varphi[x]), for any formula \varphi in the language of PA.
  • \forall x(x+0=x).
  • \forall x,y(x+Sy = S(x+y)).
  • \forall x(x \cdot 0=0).
  • \forall x,y(x \cdot Sy = (x \cdot y) + x).

PA does not require the predicate "is a natural number" because the universe of discourse of PA is just the natural numbers N. While no explicit existential quantifiers appear in the above axioms, four tacit quantifiers of that nature follow from the closure of the natural numbers under zero, successor, addition, and multiplication; and there may be implicit existential quantifiers in the axioms of induction within the \varphi.

[edit] Existence and uniqueness

A standard construction in set theory shows the existence of a Dedekind-Peano structure. First, we define the successor function; for any set a, the successor of a is S(a) := a ∪ {a}. A set A is an inductive set if it is closed under the successor function, i.e. whenever a is in A, S(a) is also in A. We now define:

  • 0 := {}
  • N := the intersection of all inductive sets containing 0
  • S := the successor function restricted to N

The set N is the set of natural numbers; it is sometimes denoted by the Greek letter ω, especially in the context of studying ordinal numbers.

The axiom of infinity guarantees the existence of an inductive set, so the set N is well-defined. The natural number system (N, 0, S) can be shown to satisfy the Peano axioms. Each natural number is then equal to the set of natural numbers less than it, so that

  • 0 := {}
  • 1 := S(0) = {0}
  • 2 := S(1) = {0,1} = {0, {0}}
  • 3 := S(2) = {0,1,2} = {0, {0}, {0, {0}}}

and so on. This construction is due to John von Neumann.

This is not the only possible construction of a Dedekind-Peano structure. For instance, if we assume the construction of the set N = {0, 1, 2,...} and successor function S above, we could also define X := {5, 6, 7,...}, x := 5, and f := successor function restricted to X. Then this is also a Dedekind-Peano structure.

5\mapsto6\mapsto7\mapsto8\mapsto\dotsb

The lambda calculus gives another construction of the natural numbers that satisfies the Peano axioms.

Two Dedekind-Peano structures (X, x, f) and (Y, y, g) are said to be isomorphic if there is a bijection φ : X → Y such that φ(x) = y and φf = gφ. It can be shown that any two Dedekind-Peano structures are isomorphic; in this sense, there is a "unique" Dedekind-Peano structure satisfying the Peano axioms. (See the categorical discussion below.)

[edit] Categorical interpretation

The Peano axioms may be interpreted in the general context of category theory. Let US1 be the category of pointed unary systems; i.e. US1 is the following category:

  • The objects of US1 are all ordered triples (X, x, f), where X is a set, x is an element of X, and f is a set map from X to itself.
  • For each (X, x, f), (Y, y, g) in US1, HomUS1((X, x, f), (Y, y, g)) = {set maps φ : X → Y | φ(x) = y and φf = gφ}
  • Composition of morphisms is the usual composition of set mappings.

The natural number system (N, 0, S) constructed above is an object in this category; it is called the natural unary system. It can be shown that the natural unary system is an initial object in US1, and so it is unique up to a unique isomorphism. This means that for any other object (X, x, f) in US1, there is a unique morphism φ : (N, 0, S) → (X, x, f). That is, that for any set X, any element x of X, and any set map f from X to itself, there is a unique set map φ : N → X such that φ(0) = x and φ(a + 1) = f(φ(a)) for all a in N. This is precisely the definition of simple recursion.

This concept can be generalised to arbitrary categories. Let C be a category with (unique) terminal object 1, and let US1(C) be the category of pointed unary systems in C; i.e. US1(C) is the following category:

  • The objects of US1(C) are all ordered triples (X, x, f), where X is an object of C, and x : 1 → X and f : X → X are morphisms in C.
  • For each (X, x, f), (Y, y, g) in US1(C), HomUS1(C)((X, x, f), (Y, y, g)) = {φ : φ is in HomC(X, Y), φx = y, and φf = gφ}
  • Composition of morphisms is the composition of morphisms in C.

Then C is said to satisfy the Dedekind-Peano axiom if there exists an initial object in US1(C). This initial object is called a natural number object in C. The simple recursion theorem is simply an expression of the fact that the natural number system (N, 0, S) is a natural number object in the category Set.

[edit] Metamathematical discussion

These axioms are given here in a second-order predicate calculus form. See first-order predicate calculus for a way to rephrase these axioms to be first-order.

Dedekind proved, in his 1888 book Was sind und was sollen die Zahlen, that any model of the second order Peano axioms is isomorphic to the natural numbers.

On the other hand, the last axiom listed above, the mathematical induction axiom, is not itself expressible in the first order language of arithmetic.

If one replaces the last axiom with the schema:

If P(0) is true; and for all x, P(x) implies P(x + 1); then P(x) is true for all x.

for each first order property P(x) (an infinite number of axioms) then although natural numbers satisfy these axioms, there are other, nonstandard models of arbitrarily large cardinality - by the compactness theorem the existence of infinite natural numbers cannot be excluded in any axiomatization; by an "upward Löwenheim-Skolem theorem", there exist models of all cardinalities. Moreover, if Dedekind's proof that the second order Peano Axioms have a unique model is viewed as a proof in a first order axiomatization of set theory such as Zermelo–Fraenkel set theory, the proof only shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism; nonstandard models of set theory may contain nonstandard models of the second order Peano Axioms.

When the axioms were first proposed, people such as Bertrand Russell agreed 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 a proof can exist that starts from just these axioms, and derives a contradiction such as P AND (NOT P), then the axioms are inconsistent, and don't really define anything. David Hilbert posed a problem of proving consistency using only finite logic as one of the problems on his famous list.

But in 1931, Kurt Gödel in his celebrated second incompleteness theorem showed such a proof cannot be given in any subsystem of Peano arithmetic. Furthermore, we can never prove that any axiom system is consistent within the system itself, if it is at least as strong as Peano's axioms, instead one must prove the consistency of the system using a different axiomatic system. Although it is widely believed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this is not clear, and 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. In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε0. As Gentzen has explained it himself: "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). Whether or not this really counts as the "finitistic proof" that Hilbert wanted is unclear: the main problem with deciding this question is that there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.

Most mathematicians assume that Peano arithmetic is consistent, although this relies on either intuition or on accepting Gentzen's proof. However, early forms of naïve set theory also intuitively looked consistent, before the inconsistencies were discovered.

Founding a mathematical system upon axioms, such as the Peano axioms for natural numbers or axiomatic set theory or Euclidean geometry is sometimes labeled the axiomatic method or axiomatics.

[edit] See also

[edit] References

  • R. Dedekind, 1888. Was sind und was sollen die Zahlen? (What are and what should the numbers be?). Braunschweig. Two English translations:
    • 1963 (1901). Essays on the Theory of Numbers. Beman, W. W., ed. and trans. Dover.
    • 1996. In From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols, Ewald, William B., ed. Oxford Uni. Press: 787-832.
  • Hatcher, William S., 1982. The Logical Foundations of Mathematics. Pergamon. A text on mathematical logic that carefully discusses the Peano axioms (called S), then derives them from several axiomatic systems of set theory.
  • Jean van Heijenoort, ed. (1967, 1976 3rd printing with corrections). From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, 3rd, Cambridge, Mass: Harvard University Press. ISBN 0-674-32449-8 (pbk.). Contains the following two papers, each preceded with valuable commentary.
  • G. Peano, Arithmetices principia, nova methodo exposita (The principles of arithmetic, presented by a new method), van Heijenoort) p. 83-97, an excerpt of his treatise wherein Peano presents his axioms, and his definitions of e.g. multiplication and division.
  • Richard Dedekind, Letter to Keferstein (1890) (van Heijenoort p. 98-103), in particular page 100 where he defines and defends his axioms of 1888.
  • Martin Davis (1974) Computability: Notes by Barry Jacobs, The Courant Institute of Mathemaatical Sciences NYU, New York.

[edit] External links

This article incorporates material from PA on PlanetMath, which is licensed under the GFDL.

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