mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...

s for the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

s presented by the 19th century

Italian Italian(s) may refer to: * Anything of, from, or related to the people of Italy over the centuries ** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom ** Italian language, a Romance language *** Regional Ita ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...

. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consisten ...

and

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...

. The need to formalize

arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...

was not well appreciated until the work of

Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...

, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881,

Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...

provided an

axiomatization In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...

of natural-number arithmetic. In 1888,

Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...

proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ( la, Arithmetices principia, nova methodo exposita). The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic". The next three axioms are first-order 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, which makes this formulation close to

second-order arithmetic In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precu ...

. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order

axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ...

Historical second-order formulation

When Peano formulated his axioms, the language of

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 In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subset ...

(∈, which comes from Peano's ε) and implication (⊃, which comes 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 ''Begriffsschrift'' (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notati ...

'' by

Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic ph ...

, published in 1879. Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of

Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely Autodidacticism, self-taught English people, English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics ...

and

Schröder Schröder ( Schroeder) is a German surname often associated with the Schröder family. Notable people with the surname include: * Arthur Schröder (1892–1986), German actor * Atze Schröder, stage name of German comedian Hubertus Albers * Bern ...

. The Peano axioms define the arithmetical properties of ''

natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

'', usually represented as a set N or

\mathbb.

The

non-logical symbol In logic, the formal languages used to create expressions consist of symbols, which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes als ...

s for the axioms consist of a constant symbol 0 and a unary function symbol ''S''. The first axiom states that the constant 0 is a natural number: Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number, while the axioms in ''

Formulario mathematico ''Formulario Mathematico'' (Latino sine flexione: ''Formulary for Mathematics'') is a book There are many editions. Here are two: * (French) Published 1901 by Gauthier-Villars, Paris. 230p.OpenLibrary OL15255022Wequality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments. The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "

successor Successor may refer to: * An entity that comes after another (see Succession (disambiguation)) Film and TV * ''The Successor'' (film), a 1996 film including Laura Girling * ''The Successor'' (TV program), a 2007 Israeli television program Mus ...

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

''S''.

Domino effect visualizing exclusion of junk term by induction axiom

Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as ''S''(0), 2 as ''S''(''S''(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. The intuitive notion that each natural number can be obtained by applying ''successor'' sufficiently often to zero requires an additional axiom, which is sometimes called the '' axiom of induction''. The induction axiom is sometimes stated in the following form: In Peano's original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section below.

Defining arithmetic operations and relations

If we use the second-order induction axiom, it is possible to define

addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or ''sum'' of ...

multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...

, and total (linear) ordering on N directly using the axioms. However, and addition and multiplication are often added as axioms. The respective functions and relations are constructed in

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...

second-order logic In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies ...

, and can be shown to be unique using the Peano axioms.

Addition

Addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or ''sum'' of ...

is a function that

maps A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Althoug ...

two natural numbers (two elements of N) to another one. It is defined

recursively Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematic ...

as: :

\begin
a + 0      &= a ,        & \textrm\\
a + S (b)  &= S (a + b). & \textrm
\end

For example: :

\begin
a + 1 &= a + S(0)     & \mbox \\
      &= S(a + 0)     & \mbox \\
      &= S(a),        & \mbox \\
\\
a + 2 &= a + S(1)     & \mbox \\
      &= S(a + 1)     & \mbox \\
      &= S(S(a))      & \mbox a + 1 = S(a) \\
\\
a + 3 &= a + S(2)     & \mbox \\
      &= S(a + 2)     & \mbox \\
      &= S(S(S(a)))    & \mbox a + 2 = S(S(a)) \\
\text & \\
\end

The structure is a

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...

with identity element 0. is also a cancellative

magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natura ...

, and thus embeddable in a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

. The smallest group embedding N is the

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

Multiplication

Similarly,

is a function mapping two natural numbers to another one. Given addition, it is defined recursively as: :

\begin
a \cdot 0 &= 0, \\
a \cdot S (b) &= a + (a \cdot b).
\end

It is easy to see that

S(0)

(or "1", in the familiar language of

decimal representation A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, ...

) is the multiplicative right identity: :

a\cdot S(0) = a + (a\cdot 0) = a + 0 = a

To show that

S(0)

is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined: *

S(0)

is the left identity of 0:

S(0)\cdot 0 = 0

. * If

S(0)

is the left identity of

a

(that is

S(0)\cdot a = a

), then

S(0)

is also the left identity of

S(a)

S(0)\cdot S(a) = S(0) + S(0)\cdot a = S(0) + a = a + S(0) = S(a + 0) = S(a)

. Therefore, by the induction axiom

S(0)

is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication is commutative and distributes over addition: :

a \cdot (b + c) = (a\cdot b) + (a\cdot c)

. Thus,

(\N, +, 0, \cdot, S(0))

is a commutative

semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs a ...

Inequalities

The usual

total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...

relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number: : For all , if and only if there exists some such that . This relation is stable under addition and multiplication: for

a, b, c \in \N

, if , then: * ''a'' + ''c'' ≤ ''b'' + ''c'', and * ''a'' · ''c'' ≤ ''b'' · ''c''. Thus, the structure 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 form that uses a stronger hypothesis, making use of the order relation "≤": : For any

predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, ...

''φ'', if :* ''φ''(0) is true, and :* for every , if ''φ''(''k'') is true for every such that , then ''φ''(''S''(''n'')) is true, :* then for every , ''φ''(''n'') is true. This form of the induction axiom, called ''strong induction'', is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are

well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well- ...

ed—every

nonempty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

of N has a

least element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an ele ...

—one can reason as follows. Let a nonempty be given and assume ''X'' has no least element. * Because 0 is the least element of N, it must be that . * For any , suppose for every , . Then , for otherwise it would be the least element of ''X''. Thus, by the strong induction principle, for every , . Thus, , which contradicts ''X'' being a nonempty subset of N. Thus ''X'' has a least element.

Models

model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...

of the Peano axioms is a triple , where N is a (necessarily infinite) set, and satisfies the axioms above.

Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. ...

proved in his 1888 book, ''The Nature and Meaning of Numbers'' (german: Was sind und was sollen die Zahlen?, i.e., “What are the numbers and what are they good for?”) that any two models of the Peano axioms (including the second-order induction axiom) are isomorphic. In particular, given two models and of the Peano axioms, there is a unique

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

satisfying :

\begin

f(0_A) &= 0_B \\
f(S_A (n)) &= S_B (f (n))
\end

and it is a bijection. This means that the second-order Peano axioms are categorical. (This is not the case with any first-order reformulation of the Peano axioms, below.)

Set-theoretic models

The Peano axioms can be derived from set theoretic constructions of the

s and axioms of set theory such as ZF. The standard construction of the naturals, due to

John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

, starts from a definition of 0 as the empty set, ∅, and an operator ''s'' on sets defined as: :

s(a) = a \cup \

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: :

\begin

0 &= \emptyset \\
1 &= s(0) = s(\emptyset) = \emptyset \cup \ = \ = \ \\
2 &= s(1) = s(\) = \ \cup \ = \ = \ \\
3 &= s(2) = s(\) = \ \cup \ = \ = \
\end

and so on. The set N together with 0 and the

successor function In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functi ...

satisfies the Peano axioms. Peano arithmetic is

equiconsistent In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not ...

with several weak systems of set theory. One such system is ZFC with the

axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing ...

replaced by its negation. Another such system consists of

general set theory General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axiom ...

(

extensionality In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...

, existence of the

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...

, and the

axiom of adjunction In mathematical set theory, the axiom of adjunction states that for any two sets ''x'', ''y'' there is a set ''w'' = ''x'' ∪ given by "adjoining" the set ''y'' to the set ''x''. : \forall x \,\forall y \,\exists w \,\forall z ...

), 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.

Interpretation in category theory

The Peano axioms can also be understood using category theory. Let ''C'' be a

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

with

terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...

1_''C'', and define the category of pointed unary systems, US₁(''C'') as follows: * The objects of US₁(''C'') are triples where ''X'' is an object of ''C'', and and are ''C''-morphisms. * A morphism ''φ'' : (''X'', 0_''X'', ''S''_''X'') → (''Y'', 0_''Y'', ''S''_''Y'') is a ''C''-morphism with and . Then ''C'' is said to satisfy the Dedekind–Peano axioms if US₁(''C'') has an initial object; this initial object is known as a

natural number object In category theory, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1, an NNO ''N'' is given by: # a global element ''z'' : 1 → ' ...

in ''C''. If is this initial object, and is any other object, then the unique map is such that :

\begin

u (0) &= 0_X, \\
u (S x) &= S_X (u x).
\end

This is precisely the recursive definition of 0_''X'' and ''S''_''X''.

Consistency

When the Peano axioms were first proposed,

Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...

and others agreed that these axioms implicitly defined what we mean by a "natural number".

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

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 twenty-three problems. In 1931,

Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...

proved his

second incompleteness theorem The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds eac ...

, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself. 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 In mathematics, logic, and computer science, a type theory is the formal system, formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theor ...

. In 1936,

Gerhard Gentzen Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died ...

gave a proof of the consistency of Peano's axioms, using

transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...

up to an ordinal called ε₀. 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 A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algor ...

describing a suitable order on the integers, or more abstractly as consisting of the finite

trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

, suitably linearly ordered). 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. A small number of philosophers and mathematicians, some of whom also advocate

ultrafinitism In the philosophy of mathematics, ultrafinitism (also known as ultraintuitionism,International Workshop on Logic and Computational Complexity, ''Logic and Computational Complexity'', Springer, 1995, p. 31. strict formalism,St. Iwan (2000),On the U ...

, 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, which are 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

\Pi_1

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").

Peano arithmetic as first-order theory

All of the Peano axioms except the ninth axiom (the induction axiom) are statements in

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quanti ...

. The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction above is second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-order ''

'' of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom. The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property). First-order axiomatizations of Peano arithmetic have another technical limitation. In second-order logic, it is possible to define the addition and multiplication operations from the successor operation, but this cannot be done in the more restrictive setting of first-order logic. Therefore, the addition and multiplication operations are directly included in the

signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...

of Peano arithmetic, and axioms are included that relate the three operations to each other. The following list of axioms (along with the usual axioms of equality), which contains six of the seven axioms of

Robinson arithmetic In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by R. M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is ...

, is sufficient for this purpose: *

\forall x \ (0 \neq  S ( x ))

\forall x, y \ (S( x ) =  S( y ) \Rightarrow x = y)

\forall x \ (x  + 0 = x )

\forall x, y \ (x + S( y ) =  S( x + y ))

\forall x \ (x \cdot 0 = 0)

\forall x, y \ (x \cdot  S ( y ) = x \cdot y + x )

In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a

recursively enumerable In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ...

and even decidable set of

axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

. For each formula in the language of Peano arithmetic, the first-order induction axiom for ''φ'' is the sentence :

\forall \bar \Bigg(\bigg(\varphi(0,\bar) \land \forall x \Big( \varphi(x,\bar)\Rightarrow\varphi(S(x),\bar)\Big)\bigg) \Rightarrow \forall x \varphi(x,\bar)\Bigg)

where

\bar

is an abbreviation for ''y''₁,...,''y''_''k''. The first-order induction schema includes every instance of the first-order induction axiom; that is, it includes the induction axiom for every formula ''φ''.

Equivalent axiomatizations

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, addition, and multiplications operations, other axiomatizations use the language of ordered semirings, including an additional order relation symbol. One such axiomatization begins with the following axioms that describe a discrete ordered semiring. #

\forall x, y, z \ ( (x + y) + z = x + (y + z) )

, i.e., addition is

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

. #

\forall x, y \ ( x + y = y + x )

, i.e., addition is

. #

\forall x, y, z \ ( (x \cdot y) \cdot z = x \cdot (y \cdot z) )

, i.e., multiplication is associative. #

\forall x, y \ ( x \cdot y = y \cdot x )

, i.e., multiplication is commutative. #

\forall x, y, z \ ( x \cdot (y + z) = (x \cdot y) + (x \cdot z) )

, i.e., multiplication distributes over addition. #

\forall x \ ( x + 0 = x \land x \cdot 0 = 0 )

, i.e., zero is an identity for addition, and an

absorbing element In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element ...

for multiplication (actually superfluous). #

\forall x \ ( x \cdot 1 = x )

, i.e., one is an identity for multiplication. #

\forall x, y, z \ ( x < y \land y < z \Rightarrow x < z )

, i.e., the '<' operator is transitive. #

\forall x \ ( \neg (x < x) )

, i.e., the '<' operator is

irreflexive In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal ...

. #

\forall x, y \ ( x < y \lor x = y \lor y < x )

, i.e., the ordering satisfies

trichotomy A trichotomy can refer to: * Law of trichotomy, a mathematical law that every real number is either positive, negative, or zero ** Trichotomy theorem, in finite group theory * Trichotomy (jazz trio), Australian jazz band, collaborators with Dan ...

. #

\forall x, y, z \ ( x < y \Rightarrow x + z < y + z )

, i.e. the ordering is preserved under addition of the same element. #

\forall x, y, z \ ( 0 < z \land x < y \Rightarrow x \cdot z < y \cdot z )

, i.e. the ordering is preserved under multiplication by the same positive element. #

\forall x, y \ ( x < y \Rightarrow \exists z \ ( x + z = y ) )

, i.e. given any two distinct elements, the larger is the smaller plus another element. #

0 < 1 \land \forall x \ ( x > 0 \Rightarrow x \ge 1 )

, i.e. zero and one are distinct and there is no element between them. In other words, 0 is

covered Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of ...

by 1, which suggests that natural numbers are discrete. #

\forall x \ ( x \ge 0 )

, i.e. zero is the minimum element. The theory defined by these axioms is known as PA⁻; the theory PA is obtained by adding the first-order induction schema. An important property of PA⁻ is that any structure

M

satisfying this theory has an initial segment (ordered by

\le

) isomorphic to

\N

. Elements in that segment are called standard elements, while other elements are called nonstandard elements.

Undecidability and incompleteness

According to

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phil ...

, the theory of PA (if consistent) is incomplete. Consequently, there are sentences of

(FOL) that are true in the standard model of PA but are not a consequence of the FOL axiomatization. Essential incompleteness already arises for theories with weaker axioms, such as

. Closely related to the above incompleteness result (via

Gödel's completeness theorem Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem applies to any first-order theory: I ...

for FOL) it follows that there is no

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

for deciding whether a given FOL sentence is a consequence of a first-order axiomatization of Peano arithmetic or not. Hence, PA is an example of an undecidable theory. Undecidability arises already for the existential sentences of PA, due to the negative answer to

Hilbert's tenth problem Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equat ...

, whose proof implies that all

computably enumerable In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ...

sets are

diophantine set In mathematics, a Diophantine equation is an equation of the form ''P''(''x''1, ..., ''x'j'', ''y''1, ..., ''y'k'') = 0 (usually abbreviated ''P''(', ') = 0) where ''P''(', ') is a polynomial with integer coefficients, where ''x''1, ..., ' ...

s, and thus definable by existentially quantified formulas (with free variables) of PA. Formulas of PA with higher quantifier rank (more quantifier alternations) than existential formulas are more expressive, and define sets in the higher levels of the

arithmetical hierarchy In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define th ...

Nonstandard models

Although the usual

s satisfy the axioms of PA, there are other models as well (called "

non-standard model In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model).Roman Kossak, 2004 ''Nonstandard Models of Arithmetic and Set Theory'' American M ...

s"); the

compactness theorem In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally ...

implies that the existence of nonstandard elements cannot be excluded in first-order logic. The upward

Löwenheim–Skolem theorem In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-orde ...

shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism. This illustrates one way the first-order system PA is weaker than the second-order Peano axioms. When interpreted as a proof within a first-order

, 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 first-order formalization of set theory. It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative as

Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem ...

in 1933 provided an explicit construction of such a

nonstandard model In model theory, a discipline within mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic c ...

. On the other hand,

Tennenbaum's theorem Tennenbaum's theorem, named for Stanley Tennenbaum who presented the theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff). Rec ...

, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is

computable Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is close ...

. This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possible

order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...

of a countable nonstandard model. Letting ''ω'' be the order type of the natural numbers, ''ζ'' be the order type of the integers, and ''η'' be the order type of the rationals, the order type of any countable nonstandard model of PA is , which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.

Overspill

A cut in a nonstandard model ''M'' is a nonempty subset ''C'' of ''M'' so that ''C'' is downward closed (''x'' < ''y'' and ''y'' ∈ ''C'' ⇒ ''x'' ∈ ''C'') and ''C'' is closed under successor. A proper cut is a cut that is a proper subset of ''M''. Each nonstandard model has many proper cuts, including one that corresponds to the standard natural numbers. However, the induction scheme in Peano arithmetic prevents any proper cut from being definable. The overspill lemma, first proved by Abraham Robinson, formalizes this fact.

Notes

References

Citations

Sources

* * ** Two English translations: *** *** * * * * * * * * Derives the Peano axioms (called S) from several

axiomatic set theories Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...

and from category theory. * * * * * * * * * * * Derives the Peano axioms from ZFC * * ** Contains translations of the following two papers, with valuable commentary: *** *** * *

External links

* Includes a discussion of Poincaré's critique of the Peano's axioms. * * * * Commentary on Dedekind's work. {{PlanetMath attribution, urlname=pa, title=PA 1889 introductions Mathematical axioms Formal theories of arithmetic Logic in computer science Mathematical logic hu:Giuseppe Peano#A természetes számok Peano-axiómái