TheInfoList

In
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 sys ...
, 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 truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' 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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s presented by the 19th century
Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Italian language, a Romance language *** Regional Italian, regional variants of the ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

. 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 devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

is
consistent In classical deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion. Deductive reasoning goes in the same direction as that of the conditiona ...
and complete. The need to formalize
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
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 linguistics, linguist and now also as a mathematics, mathematician. He was also a physics, physicist, gener ...
, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and
induction Induction may refer to: Philosophy * Inductive reasoning, in logic, inferences from particular cases to the general case Biology and chemistry * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induction period, the t ...
. In 1881,
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, ian, mathematician and scientist who is sometimes known as "the father of ". He was known as a somewhat unusual character. Educated as a chemist an ...

provided an
axiomatization In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
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. 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 schemaIn 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 prop ...
.

Formulation

When Peano formulated his axioms, the language of
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 sys ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(∈, 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 notatio ...
'' by
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
, 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 self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork Universi ...

and Schröder. 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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

'', usually represented as a set N or $\mathbb.$ The
non-logical symbolIn logic, the formal languages used to create expressions consist of symbol (formal), symbols, which can be broadly divided into logical constants, constants and Variable (mathematics), variables. The constants of a language can further be divided in ...
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'' include zero. The next four axioms describe the equality 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 is someone who, or something which succeeds or comes after (see success (disambiguation), success and Succession (disambiguation), succession) Film and TV * The Successor (film), ''The Successor'' (film), a 1996 film including Laura Girli ...
"
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
''S''. 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 Image:Dominoeffect.png, Mathematical induction can be informally illustrated by reference to the sequential effect of falling Domino effect, dominoes. Mathematical induction is a mathematical proof technique. It is essentially used to prove that ...
''. 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.

Arithmetic

The Peano axioms can be augmented with the operations of
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

and
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 arithmetic, elementary Operation (mathematics), mathematical operations ...

and the usual total (linear) ordering on N. The respective functions and relations are constructed in
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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, i ...
or
second-order logic In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
, and can be shown to be unique using the Peano axioms.

Addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...
is a function that
maps A map is a symbol A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meani ...
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 mathematics ...

as: : $\begin a + 0 &= a , & \textrm\\ a + S \left(b\right) &= S \left(a + b\right). & \textrm \end$ For example: : $\begin a + 1 &= a + S\left(0\right) & \mbox \\ &= S\left(a + 0\right) & \mbox \\ &= S\left(a\right), & \mbox \\ \\ a + 2 &= a + S\left(1\right) & \mbox \\ &= S\left(a + 1\right) & \mbox \\ &= S\left(S\left(a\right)\right) & \mbox a + 1 = S\left(a\right) \\ \\ a + 3 &= a + S\left(2\right) & \mbox \\ &= S\left(a + 2\right) & \mbox \\ &= S\left(S\left(S\left(a\right)\right)\right) & \mbox a + 2 = S\left(S\left(a\right)\right) \\ \text & \\ \end$ The
structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
is a
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
with identity element 0. is also a
cancellative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
magma Magma () is the molten or semi-molten natural material from which all igneous rock Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the others ...
, and thus embeddable in a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
. The smallest group embedding N is the
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s.

Multiplication

Similarly,
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 arithmetic, elementary Operation (mathematics), mathematical operations ...

is a function mapping two natural numbers to another one. Given addition, it is defined recursively as: : $\begin a \cdot 0 &= 0, \\ a \cdot S \left(b\right) &= a + \left(a \cdot b\right). \end$ It is easy to see that $S\left(0\right)$ (or "1", in the familiar language of
decimal representation A decimal representation of a non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive n ...
) is the multiplicative right identity: :$a\cdot S\left(0\right) = a + \left(a\cdot 0\right) = a + 0 = a$ To show that $S\left(0\right)$ is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined: * $S\left(0\right)$ is the left identity of 0: $S\left(0\right)\cdot 0 = 0$. * If $S\left(0\right)$ is the left identity of $a$ (that is $S\left(0\right)\cdot a = a$), then $S\left(0\right)$ is also the left identity of $S\left(a\right)$: $S\left(0\right)\cdot S\left(a\right) = S\left(0\right) + S\left(0\right)\cdot a = S\left(0\right) + a = a + S\left(0\right) = S\left(a + 0\right) = S\left(a\right)$. Therefore, by the induction axiom $S\left(0\right)$ is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication is commutative and distributes over addition: : $a \cdot \left(b + c\right) = \left(a\cdot b\right) + \left(a\cdot c\right)$. Thus, $\left(\N, +, 0, \cdot, S\left(0\right)\right)$ is a commutative
semiring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
.

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 (mathematics), set X, which satisfies the following for all a, b and c in X: # a \ ...
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: Computer science *Syntactic predicate (in parser technology) guidelines the parser process Linguistics *Predicate (grammar), a grammatical component of a sentence Philosophy and logic * Predication (philo ...
''φ'', if :* ''φ''(0) is true, and :* for every , if implies that ''φ''(''k'') is true, 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 (mathematics), 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 ...
ed—every
nonempty In mathematics, the empty set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by includ ...

subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of N has a
least element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
—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 ''X'' being a nonempty subset of N. Thus ''X'' has a least element.

First-order theory of arithmetic

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 Quantificat ...
. The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction is in second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers), but it can be transformed into a first-order ''
axiom schemaIn 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 prop ...
'' 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 handwritten (and often 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 intent. The writer of a s ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, is sufficient for this purpose: * $\forall x \ \left(0 \neq S \left( x \right)\right)$ * $\forall x, y \ \left(S\left( x \right) = S\left( y \right) \Rightarrow x = y\right)$ * $\forall x \ \left(x + 0 = x \right)$ * $\forall x, y \ \left(x + S\left( y \right) = S\left( x + y \right)\right)$ * $\forall x \ \left(x \cdot 0 = 0\right)$ * $\forall x, y \ \left(x \cdot S \left( y \right) = x \cdot y + x \right)$ In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a
recursively enumerable In computability theory, traditionally called recursion theory, a set ''S'' of natural numbers is called recursively enumerable, computably enumerable, semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is a ...
set of
axioms An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
. For each formula in the language of Peano arithmetic, the first-order induction axiom for ''φ'' is the sentence :$\forall \bar \Bigg\left(\bigg\left(\varphi\left(0,\bar\right) \land \forall x \Big\left( \varphi\left(x,\bar\right)\Rightarrow\varphi\left(S\left(x\right),\bar\right)\Big\right)\bigg\right) \Rightarrow \forall x \varphi\left(x,\bar\right)\Bigg\right)$ 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. # $\forall x, y, z \ \left( \left(x + y\right) + z = x + \left(y + z\right) \right)$, i.e., addition is
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. # $\forall x, y \ \left( x + y = y + x \right)$, i.e., addition is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. # $\forall x, y, z \ \left( \left(x \cdot y\right) \cdot z = x \cdot \left(y \cdot z\right) \right)$, i.e., multiplication is associative. # $\forall x, y \ \left( x \cdot y = y \cdot x \right)$, i.e., multiplication is commutative. # $\forall x, y, z \ \left( x \cdot \left(y + z\right) = \left(x \cdot y\right) + \left(x \cdot z\right) \right)$, i.e., multiplication distributes over addition. # $\forall x \ \left( x + 0 = x \land x \cdot 0 = 0 \right)$, i.e., zero is an
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
absorbing elementIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
for multiplication (actually superfluous). # $\forall x \ \left( x \cdot 1 = x \right)$, i.e., one is an
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
for multiplication. # $\forall x, y, z \ \left( x < y \land y < z \Rightarrow x < z \right)$, i.e., the '<' operator is
transitive Transitivity or transitive may refer to: Grammar * Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects * Transitive verb, a verb which takes an object * Transitive case, a grammatical case to mark arg ...
. # $\forall x \ \left( \neg \left(x < x\right) \right)$, i.e., the '<' operator is
irreflexive In mathematics, a homogeneous binary relation ''R'' over a set (mathematics), set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation "equality (mathematics), is equal to" on the se ...
. # $\forall x, y \ \left( x < y \lor x = y \lor y < x \right)$, i.e., the ordering satisfies trichotomy. # $\forall x, y, z \ \left( x < y \Rightarrow x + z < y + z \right)$, i.e. the ordering is preserved under addition of the same element. # $\forall x, y, z \ \left( 0 < z \land x < y \Rightarrow x \cdot z < y \cdot z \right)$, i.e. the ordering is preserved under multiplication by the same positive element. # $\forall x, y \ \left( x < y \Rightarrow \exists z \ \left( x + z = y \right) \right)$, i.e. given any two distinct elements, the larger is the smaller plus another element. # $0 < 1 \land \forall x \ \left( x > 0 \Rightarrow x \ge 1 \right)$, 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, science and technology * A covering, usually - but not necessarily - one that completely closes the object ** Cover (philately), generic term for envelope or package ** Housing (engineering), an exterior ...
by 1, which suggests that natural numbers are discrete. # $\forall x \ \left( x \ge 0 \right)$, 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.

Models

A
model In general, 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. ...
of the Peano axioms is a triple , where N is a (necessarily infinite) set, and satisfies the axioms above. Dedekind 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 mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

. In particular, given two models and of the Peano axioms, there is a unique
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
satisfying : $\begin f\left(0_A\right) &= 0_B \\ f\left(S_A \left(n\right)\right) &= S_B \left(f \left(n\right)\right) \end$ and it is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. This means that the second-order Peano axioms are categorical. This is not the case with any first-order reformulation of the Peano axioms, however.

Set-theoretic models

The Peano axioms can be derived from set theoretic constructions of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
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 Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

, starts from a definition of 0 as the empty set, ∅, and an operator ''s'' on sets defined as: : $s\left(a\right) = 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: : $\begin 0 &= \emptyset \\ 1 &= s\left(0\right) = s\left(\emptyset\right) = \emptyset \cup \ = \ = \ \\ 2 &= s\left(1\right) = s\left(\\right) = \ \cup \ = \ = \ \\ 3 &= s\left(2\right) = s\left(\\right) = \ \cup \ = \ = \ \end$ and so on. The set N together with 0 and the
successor function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
satisfies the Peano axioms. Peano arithmetic is equiconsistent 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...
replaced by its negation. Another such system consists of
general set theoryGeneral set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Zermelo set theory, Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include th ...
(Axiom of extensionality, extensionality, existence of the empty set, and the general set theory, 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.

Interpretation in category theory

The Peano axioms can also be understood using category theory. Let ''C'' be a category (mathematics), category with terminal object 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 ''C''. If is this initial object, and is any other object, then the unique map is such that : $\begin u \left(0\right) &= 0_X, \\ u \left(S x\right) &= S_X \left(u x\right). \end$ This is precisely the recursive definition of 0''X'' and ''S''''X''.

Nonstandard models

Although the usual
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s satisfy the axioms of #Equivalent axiomatizations, PA, there are other models as well (called "non-standard models"); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order 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 (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
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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, i ...
, such as Zermelo–Fraenkel set theory, 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 in 1933 provided an explicit construction of such a Non-standard model of arithmetic, nonstandard model. On the other hand, Tennenbaum's theorem, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable function, computable. 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 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.

Consistency

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 Finitism, finitistic methods as the Hilbert's second problem, second of his Hilbert's problems, twenty-three problems. In 1931, Kurt Gödel proved his second incompleteness theorem, 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 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an Ordinal number, ordinal called epsilon zero, 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 describing a suitable order on the integers, or more abstractly as consisting of the finite Tree (set theory), trees, 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 consistency proof, Gentzen's proof. 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 Partial function#Total function, 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").

* Foundations of mathematics * Frege's theorem * Goodstein's theorem * Neo-logicism * Non-standard model of arithmetic * Paris–Harrington theorem * Presburger arithmetic *
Robinson arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
* Second-order arithmetic * Typographical Number Theory

References

Sources

* * ** Two English translations: *** *** * * * * * * * * Derives the Peano axioms (called S) from several axiomatic set theories and from category theory. * * * * * * * * * Derives the Peano axioms from ZFC * * ** Contains translations of the following two papers, with valuable commentary: *** *** * *