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| In [[mathematical logic]], the '''Peano axioms''', also known as the '''Dedekind–Peano axioms''' or the '''Peano postulates''', are a set of [[axiom]]s for the [[natural number]]s presented by the 19th century [[Italian people|Italian]] [[mathematician]] [[Giuseppe Peano]]. These axioms have been used nearly unchanged in a number of [[metamathematics|metamathematical]] investigations, including research into fundamental questions of [[consistency proof|consistency]] and [[completeness]] of [[number theory]].
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| The need for formalism in [[arithmetic]] was not well appreciated until the work of [[Hermann Grassmann]], who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the [[successor function|successor operation]] and [[mathematical induction|induction]].<ref>Grassmann 1861</ref> In 1881, [[Charles Sanders Peirce]] provided an [[Axiomatic system#Axiomatization|axiomatization]] of natural-number arithmetic.<ref>Peirce 1881; also see Shields 1997</ref> In 1888, [[Richard Dedekind]] proposed a collection of axioms about the numbers, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ({{lang-la|Arithmetices principia, nova methodo exposita}}).
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| The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set "number". The next four are general statements about [[Equality (mathematics)|equality]]; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".<ref>van Heijenoort 1967:94</ref> The next three axioms are [[first-order logic|first-order]] statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a [[second-order logic|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 arithmetic#Induction and comprehension schema|second-order induction]] axiom with a first-order [[axiom schema]].
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| == The axioms ==
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| {| style = "float:right"
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| | [[File:Domino effect visualizing exclusion of junk term by induction axiom.jpg|thumb|The set of natural numbers can be illustrated by the infinite chain of light wood domino pieces, their first one corresponding to zero, and each piece facing its top side towards its successor. However, the Peano axioms 1-8 are also fulfilled by the incontiguous structure consisting of both light and dark wood pieces. The induction axiom, 9, corresponds to the requirement that if the first light wood domino piece (0) is overthrown, then each piece will eventually fall ("[[domino effect]]"); this is satisfied only in the absence of the dark pieces.]]
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| |}
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| When Peano formulated his axioms, the language of [[mathematical logic]] was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for [[Element (mathematics)|set membership]] (∈, which comes from Peano's ε) and [[logical implication|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]] by [[Gottlob Frege]], published in 1879.<ref>Van Heijenoort 1967, p. 2</ref> Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of [[George Boole|Boole]] and [[Ernst Schröder|Schröder]].<ref>Van Heijenoort 1967, p. 83</ref>
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| The Peano axioms define the arithmetical properties of ''[[natural numbers]]'', usually represented as a [[Set (mathematics)|set]] ''N'' or <math>\mathbb{N}.</math> The [[signature (mathematical logic)|signature]] (a formal language's [[non-logical symbol]]s) for the axioms includes a constant symbol 0 and a unary function symbol ''S''.
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| The constant 0 is assumed to be a natural number:
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| #0 is a natural number.
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| The next four axioms describe the [[equality (mathematics)|equality]] [[relation (mathematics)|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.<ref>Van Heijenoort 1967, p. 83</ref>
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| #<li value="2"> For every natural number ''x'', ''x'' = ''x''. That is, equality is [[reflexive relation|reflexive]].
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| #For all natural numbers ''x'' and ''y'', if ''x'' = ''y'', then ''y'' = ''x''. That is, equality is [[symmetric relation|symmetric]].
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| #For all natural numbers ''x'', ''y'' and ''z'', if ''x'' = ''y'' and ''y'' = ''z'', then ''x'' = ''z''. That is, equality is [[transitive relation|transitive]].
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| #For all ''a'' and ''b'', if ''a'' is a natural number and ''a'' = ''b'', then ''b'' is also a natural number. That is, the natural numbers are [[closure (mathematics)|closed]] under equality.
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| The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor" [[function (mathematics)|function]] ''S''.
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| #<li value="6">For every natural number ''n'', ''S''(''n'') is a natural number.
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| Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the [[identity element|additive identity]] in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1 and 6 define a [[unary numeral system|unary]] representation of the natural numbers: the number 1 can be defined as ''S''(0), 2 as ''S''(''S''(0)) (which is also ''S''(1)), and, in general, any natural number ''n'' as the result of ''n''-fold application of ''S'' to 0, denoted as ''S''<sup>''n''</sup>(0). The next two axioms define the properties of this representation.
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| #<li value="7">For every natural number ''n'', ''S''(''n'') = 0 is false. That is, there is no natural number whose successor is 0.
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| #For all natural numbers ''m'' and ''n'', if ''S''(''m'') = ''S''(''n''), then ''m'' = ''n''. That is, ''S'' is an [[injective function|injection]].
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| Axioms 1, 6, 7 and 8 imply that the set of natural numbers contains the distinct elements 0, ''S''(0), ''S''(''S''(0)), and furthermore that {0, ''S''(0), ''S''(''S''(0)), …} ⊆ ''N''. This shows that the set of natural numbers is infinite. However, to show that ''N'' = {0, ''S''(0), ''S''(''S''(0)), …}, it must be shown that ''N'' ⊆ {0, ''S''(0), ''S''(''S''(0)), …}; i.e., it must be shown that every natural number is included in {0, ''S''(0), ''S''(''S''(0)), …}. To do this however requires an additional axiom, which is sometimes called the ''axiom of induction''. This axiom provides a method for reasoning about the set of all natural numbers.
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| <ol>
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| <li value="9">
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| If ''K'' is a set such that:
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| * ''0'' is in ''K'', and
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| * for every natural number ''n'', if ''n'' is in ''K'', then ''S''(''n'') is in ''K'',
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| then ''K'' contains every natural number.
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| </li>
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| </ol>
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| The induction axiom is sometimes stated in the following form:
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| <ol>
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| <li value="9">If φ is a unary [[predicate (mathematics)|predicate]] such that:
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| * φ(0) is true, and
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| * for every natural number ''n'', if φ(''n'') is true, then φ(''S''(''n'')) is true,
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| then φ(''n'') is true for every natural number ''n''.
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| </li>
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| </ol>
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| In Peano's original formulation, the induction axiom is a [[second-order logic|second-order axiom]]. It is now common to replace this second-order principle with a weaker [[first-order logic|first-order]] induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section [[#Models|Models]] below.
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| == Arithmetic ==
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| The Peano axioms can be augmented with the operations of [[addition]] and [[multiplication]] and the usual [[total order|total (linear) ordering]] on ''N''. The respective functions and relations are constructed in [[second-order logic]], and are shown to be unique using the Peano axioms.
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| === Addition ===
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| [[Addition in N|Addition]] is the function + : ''N'' × ''N'' → ''N'' (written in the usual [[infix notation]], [[map (mathematics)|mapping]] two elements of ''N'' to another element of ''N''), defined [[recursion|recursively]] as:
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| :<math>\begin{align}
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| a + 0 &= a ,\\
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| a + S (b) &= S (a + b).
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| \end{align}</math>
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| For example,
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| :''a'' + 1 = ''a'' + ''S''(0) = ''S''(''a'' + 0) = ''S''(''a'').
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| The [[Structure (mathematical logic)|structure]] (''N'', +) is a [[commutative]] [[semigroup]] with identity element 0. (''N'', +) is also a [[cancellation property|cancellative]] [[magma (algebra)|magma]], and thus [[embedding|embeddable]] in a [[group (mathematics)|group]]. The smallest group embedding ''N'' is the [[integer]]s.
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| === Multiplication ===
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| Given addition, [[multiplication]] is the function · : ''N'' × ''N'' → ''N'' defined recursively as:
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| :<math>\begin{align}
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| a \cdot 0 &= 0, \\
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| a \cdot S (b) &= a + (a \cdot b).
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| \end{align}</math>
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| It is easy to see that setting b equal to 0 yields the multiplicative [[identity element|identity]]:
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| :''a'' · 1 = ''a'' · ''S''(0) = ''a'' + (''a'' · 0) = ''a'' + 0 = ''a''
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| Moreover, multiplication [[distributive law|distributes over]] addition:
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| :''a'' · (''b'' + ''c'') = (''a'' · ''b'') + (''a'' · ''c'').
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| Thus, (''N'', +, 0, ·, 1) is a commutative [[semiring]].
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| === Inequalities ===
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| The usual [[total order]] relation ≤ : ''N'' × ''N'' can be defined as follows, assuming 0 is a natural number:
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| :For all ''a'', ''b'' ∈ ''N'', ''a'' ≤ ''b'' if and only if there exists some ''c'' ∈ ''N'' such that ''a'' + ''c'' = ''b''.
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| This relation is stable under addition and multiplication: for <math> a, b, c \in N </math>, if ''a'' ≤ ''b'', then:
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| * ''a'' + ''c'' ≤ ''b'' + ''c'', and
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| * ''a'' · ''c'' ≤ ''b'' · ''c''.
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| Thus, the structure (''N'', +, ·, 1, 0, ≤) is an [[ordered ring|ordered semiring]]; because there is no natural number between 0 and 1, it is a discrete ordered semiring. The axiom of induction is sometimes stated in the following ''strong'' form, making use of the ≤ order:
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| :For any [[predicate (mathematics)|predicate]] φ, if
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| :* φ(0) is true, and
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| :* for every ''n'', ''k'' ∈ ''N'', if ''k'' ≤ ''n'' implies ''φ''(''k'') is true, then φ(''S''(''n'')) is true,
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| :then for every ''n'' ∈ ''N'', φ(''n'') is true.
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| This form of the induction axiom is a simple consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are [[well-order]]ed—every [[empty set|nonempty]] [[subset]] of ''N'' has a [[least element]]—one can reason as follows. Let a nonempty ''X'' ⊆ ''N'' be given and assume ''X'' has no least element.
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| * Because 0 is the least element of ''N'', it must be that 0 ∉ ''X''.
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| * For any ''n'' ∈ ''N'', suppose for every ''k'' ≤ ''n'', ''k'' ∉ ''X''. Then ''S''(''n'') ∉ ''X'', for otherwise it would be the least element of ''X''.
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| Thus, by the strong induction principle, for every ''n'' ∈ ''N'', ''n'' ∉ ''X''. Thus, ''X'' ∩ ''N'' = ∅, which [[contradiction|contradicts]] ''X'' being a nonempty subset of ''N''. Thus ''X'' has a least element.
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| == First-order theory of arithmetic ==
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| [[First-order logic|First-order]] theories are often better than [[second-order logic|second order]] theories for [[model theory|model]] or [[proof theory|proof theoretic]] analysis. All of the Peano axioms except the ninth axiom (the induction axiom) are statements in [[first-order logic]]. The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The second-order axiom of induction can be transformed into a weaker '''first-order induction [[axiom schema|schema]]'''.
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| First-order axiomatizations of Peano arithmetic have an important limitation, however. 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 of Peano arithmetic, and axioms are included that relate the three operations to each other.
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| The following list of axioms (along with the usual axioms of equality) is sufficient for this purpose:<ref>Mendelson 1997:155</ref>
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| * ∀''x''<sub>1</sub>∈''N''. 0 ≠ ''S''(''x''<sub>1</sub>)
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| * ∀''x''<sub>1</sub>,''x''<sub>2</sub>∈''N''. ''S''(''x''<sub>1</sub>) = ''S''(''x''<sub>2</sub>) ⇒ ''x''<sub>1</sub> = ''x''<sub>2</sub>
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| * ∀''x''<sub>1</sub>∈''N''. ''x''<sub>1</sub> + 0 = ''x''<sub>1</sub>
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| * ∀''x''<sub>1</sub>,''x''<sub>2</sub>∈''N''. ''x''<sub>1</sub> + ''S''(''x''<sub>2</sub>) = ''S''(''x''<sub>1</sub> + ''x''<sub>2</sub>)
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| * ∀''x''<sub>1</sub>∈''N''. ''x''<sub>1</sub> ⋅ 0 = 0
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| * ∀''x''<sub>1</sub>,''x''<sub>2</sub>∈''N''. ''x''<sub>1</sub> ⋅ ''S''(''x''<sub>2</sub>) = ''x''<sub>1</sub> ⋅ ''x''<sub>2</sub> + ''x''<sub>1</sub>
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| In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a [[Countable set|countably infinite]] set of [[axioms]]. For each formula φ(''x'',''y''<sub>1</sub>,...,''y''<sub>''k''</sub>) in the language of Peano arithmetic, the '''first-order induction axiom''' for φ is the sentence
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| :<math>\forall \bar{y} (\varphi(0,\bar{y}) \land \forall x ( \varphi(x,\bar{y})\Rightarrow\varphi(S(x),\bar{y})) \Rightarrow \forall x \varphi(x,\bar{y}))</math>
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| where <math>\bar{y}</math> is an abbreviation for ''y''<sub>1</sub>,...,''y''<sub>''k''</sub>. 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.
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| This schema avoids quantification over sets of natural numbers, which is impossible in first-order logic. For instance, it is not possible in first-order logic to say that any set of natural numbers containing 0 and closed under successor is the entire set of natural numbers. What can be expressed is that any [[Structure (mathematical logic)|definable]] set of natural numbers has this property. Because it is not possible to quantify over definable subsets explicitly with a single axiom, the induction schema includes one instance of the induction axiom for every definition of a subset of the naturals.
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| === Equivalent axiomatizations ===
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| 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 ring|ordered semirings]], including an additional order relation symbol. One such axiomatization begins with the following axioms that describe a discrete ordered semiring.<ref>Kaye 1991, pp. 16–18</ref>
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| <!-- These axioms are taken directly from Kaye 1991. Please don't "tweak" them, add additional axioms, or remove axioms without discussion on the talk page. -->
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| # <math>\forall x, y, z \in N</math>. <math>(x + y) + z = x + (y + z)</math>, i.e., addition is [[associative law|associative]].
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| # <math>\forall x, y \in N</math>. <math>x + y = y + x</math>, i.e., addition is [[commutative law|commutative]].
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| # <math>\forall x, y, z \in N</math>. <math>(x \cdot y) \cdot z = x \cdot (y \cdot z)</math>, i.e., multiplication is associative.
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| # <math>\forall x, y \in N</math>. <math>x \cdot y = y \cdot x</math>, i.e., multiplication is commutative.
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| # <math>\forall x, y, z \in N</math>. <math>x \cdot (y + z) = (x \cdot y) + (x \cdot z)</math>, i.e., the [[distributive law]].
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| # <math>\forall x \in N</math>. <math>x + 0 = x \and x \cdot 0 = 0</math>, i.e., zero is the identity element for addition.
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| # <math>\forall x \in N</math>. <math>x \cdot 1 = x</math>, i.e., one is the identity element for multiplication.
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| # <math>\forall x, y, z \in N</math>. <math>x < y \and y < z \Rightarrow x < z</math>, i.e., the '<' operator is [[Transitive relation|transitive]].
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| # <math>\forall x \in N</math>. <math>\neg (x < x)</math>, i.e., the '<' operator is [[Reflexive relation|irreflexive]].
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| # <math>\forall x, y \in N</math>. <math>x < y \or x = y \or y < x</math>.
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| # <math>\forall x, y, z \in N</math>. <math>x < y \Rightarrow x + z < y + z</math>.
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| # <math>\forall x, y, z \in N</math>. <math>0 < z \and x < y \Rightarrow x \cdot z < y \cdot z</math>.
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| # <math>\forall x, y \in N</math>. <math>x < y \Rightarrow \exists z \in N</math>. <math>x + z = y</math>.
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| # <math>0 < 1 \and \forall x \in N</math>. <math>x > 0 \Rightarrow x \geq 1</math>.
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| # <math>\forall x \in N</math>. <math>x \geq 0</math>.
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| The theory defined by these axioms is known as '''PA<sup>–</sup>'''; PA is obtained by adding the first-order induction schema.
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| An important property of PA<sup>–</sup> is that any structure ''M'' satisfying this theory has an initial segment (ordered by ≤) isomorphic to ''N''. Elements of ''M'' \ ''N'' are known as '''nonstandard''' elements.
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| == Models ==
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| A [[model theory|model]] of the Peano axioms is a triple (''N'', 0, ''S''), where ''N'' is a (necessarily infinite) set, 0 ∈ ''N'' and ''S'' : ''N'' → ''N'' satisfies the axioms above. [[Richard Dedekind|Dedekind]] proved in his 1888 book, ''What are numbers and what should they be'' ({{lang-de|Was sind und was sollen die Zahlen}}) that any two models of the Peano axioms (including the second-order induction axiom) are [[isomorphism|isomorphic]]. In particular, given two models (''N''<sub>''A''</sub>, 0<sub>''A''</sub>, ''S''<sub>''A''</sub>) and (''N''<sub>''B''</sub>, 0<sub>''B''</sub>, ''S''<sub>''B''</sub>) of the Peano axioms, there is a unique [[homomorphism]] ''f'' : ''N''<sub>''A''</sub> → ''N''<sub>''B''</sub> satisfying
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| :<math>\begin{align}
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| f(0_A) &= 0_B \\
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| f(S_A (n)) &= S_B (f (n))
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| \end{align}</math>
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| and it is a [[bijective function|bijection]]. The second-order Peano axioms are thus [[Morley's categoricity theorem|categorical]]; this is not the case with any first-order reformulation of the Peano axioms, however.
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| === Nonstandard models ===
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| Although the usual [[natural number]]s satisfy the axioms of PA, there are other '''[[non-standard model]]s''' as well; 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.
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| When interpreted as a proof within a first-order set theory, 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.
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| It is natural to ask whether a countable nonstandard model can be explicitly constructed. [[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]].<ref>Kaye 1991, sec. 11.3</ref> This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. However, 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 ω + ζ·η, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.
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| === Set-theoretic models ===
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| {{Main|Set-theoretic definition of natural numbers}}
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| The Peano axioms can be derived from [[set theory|set theoretic]] constructions of the [[natural number]]s and axioms of set theory such as the [[Zermelo–Fraenkel set theory|ZF]].<ref>Suppes 1960; Hatcher 1982</ref> The standard construction of the naturals, due to [[John von Neumann]], starts from a definition of 0 as the empty set, ∅, and an operator ''s'' on sets defined as:
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| :''s''(''a'') = ''a'' ∪ { ''a'' }.
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| The set of natural numbers '''N''' is defined as the intersection of all sets [[closure (mathematics)|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:
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| :<math>\begin{align}
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| 0 &= \emptyset \\
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| 1 &= s(0) = s(\emptyset) = \emptyset \cup \{ \emptyset \} = \{ \emptyset \} = \{ 0 \} \\
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| 2 &= s(1) = s(\{ 0 \}) = \{ 0 \} \cup \{ \{ 0 \} \} = \{ 0 , \{ 0 \} \} = \{ 0, 1 \} \\
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| 3 &= ... = \{ 0, 1, 2 \}
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| \end{align}</math>
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| and so on. The set '''N''' together with 0 and the successor function ''s'' : '''N''' → '''N''' satisfies the Peano axioms. <!-- dubious: The induction axiom is proven using the [[axiom of infinity]] of set theory.-->
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| Peano arithmetic is [[equiconsistent]] with several weak systems of set theory.<ref>Tarski & Givant 1987, sec. 7.6</ref> One such system is ZFC with the axiom of infinity replaced by its negation. Another such system consists of [[general set theory]] ([[extensionality]], existence of the [[empty set]], and the [[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.
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| === Interpretation in category theory ===
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| The Peano axioms can also be understood using [[category theory]]. Let ''C'' be a [[category (mathematics)|category]] with [[terminal object]] 1<sub>''C''</sub>, and define the category of '''pointed unary systems''', US<sub>1</sub>(''C'') as follows:
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| * The objects of US<sub>1</sub>(''C'') are triples (''X'', 0<sub>''X''</sub>, ''S''<sub>''X''</sub>) where ''X'' is an object of ''C'', and 0<sub>''X''</sub> : 1<sub>''C''</sub> → ''X'' and ''S''<sub>''X''</sub> : ''X'' → ''X'' are ''C''-morphisms.
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| * A morphism φ : (''X'', 0<sub>''X''</sub>, ''S''<sub>''X''</sub>) → (''Y'', 0<sub>''Y''</sub>, ''S''<sub>''Y''</sub>) is a ''C''-morphism φ : ''X'' → ''Y'' with φ 0<sub>''X''</sub> = 0<sub>''Y''</sub> and φ ''S''<sub>''X''</sub> = ''S''<sub>''Y''</sub> φ.
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| Then ''C'' is said to satisfy the Dedekind–Peano axioms if US<sub>1</sub>(''C'') has an initial object; this initial object is known as a [[natural number object]] in ''C''. If (''N'', 0, ''S'') is this initial object, and (''X'', 0<sub>''X''</sub>, ''S''<sub>''X''</sub>) is any other object, then the unique map ''u'' : (''N'', 0, ''S'') → (''X'', 0<sub>''X''</sub>, ''S''<sub>''X''</sub>) is such that
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| :<math>\begin{align}
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| u 0 &= 0_X, \\
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| u (S x) &= S_X (u x).
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| \end{align}</math>
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| This is precisely the recursive definition of 0<sub>''X''</sub> and ''S''<sub>''X''</sub>.
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| == Consistency ==
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| {{Main|Hilbert's second problem}}
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| 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]].<ref>Hilbert 1900</ref> In 1931, [[Kurt Gödel]] proved his [[Gödel's incompleteness theorem|second incompleteness theorem]], which shows that such a consistency proof cannot be formalized within Peano arithmetic itself.<ref>Godel 1931</ref>
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| 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]].<ref>Godel 1958</ref> In 1936, [[Gerhard Gentzen]] gave a proof of the consistency of Peano's axioms, using [[transfinite induction]] up to an ordinal called [[epsilon zero|ε<sub>0</sub>]].<ref>Gentzen 1936</ref> Gentzen explained: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε<sub>0</sub> can be encoded in terms of finite objects (for example, as a [[Turing machine]] describing a suitable order on the integers, or more abstractly as consisting of the finite trees, suitably linearly ordered). 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.
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| 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]]. The small number of mathematicians who advocate [[ultrafinitism]] reject Peano's axioms because the axioms require an infinite set of natural numbers.
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| ==See also==
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| * [[Foundations of mathematics]]
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| * [[Gentzen's consistency proof]]
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| * [[Goodstein's theorem]]
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| * [[Paris–Harrington theorem]]
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| * [[Presburger arithmetic]]
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| * [[Robinson arithmetic]]
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| * [[Second-order arithmetic]]
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| * [[Non-standard model of arithmetic]]
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| * [[Set-theoretic definition of natural numbers]]
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| * [[Frege's theorem]]
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| == Footnotes ==
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| {{reflist|2}}
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| == References ==
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| * [[Martin Davis]], 1974. ''Computability''. Notes by Barry Jacobs. [[Courant Institute of Mathematical Sciences]], [[New York University]].
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| * {{cite book| author=Richard Dedekind|authorlink=Richard Dedekind| title=Was sind und was sollen die Zahlen? (What are and what should the numbers be?)| year=1888| publisher=Vieweg| url=http://digisrv-1.biblio.etc.tu-bs.de:8080/docportal/servlets/MCRFileNodeServlet/DocPortal_derivate_00005731/V.C.125.pdf;jsessionid=FB796823C290377DAA1A50426E5B7E53| accessdate=31 October 2013}}. Two English translations:
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| **1963 (1901). [http://www.gutenberg.org/files/21016/21016-pdf.pdf ''Essays on the Theory of Numbers'']. Beman, W. W., ed. and trans. Dover.
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| **1996. In ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics'', 2 vols, Ewald, William B., ed. Oxford University Press: 787–832.
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| * [[Gerhard Gentzen|Gentzen, G.]], 1936, ''Die Widerspruchsfreiheit der reinen Zahlentheorie.'' ''Mathematische Annalen'' 112: 132–213. Reprinted in English translation in his 1969 ''Collected works'', M. E. Szabo, ed. Amsterdam: North-Holland.
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| * {{cite journal| author=Kurt Gödel|authorlink=Kurt Gödel| title=Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I| journal=Monatshefte für Mathematik| year=1931| volume=38| pages=173–198| url=http://www.w-k-essler.de/pdfs/goedel.pdf}}. See [[On Formally Undecidable Propositions of Principia Mathematica and Related Systems]] for details on English translations.
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| *--------, 1958, "Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes," ''Dialectica'' 12: 280–87. Reprinted in English translation in 1990. Gödel's ''Collected Works'', Vol II. [[Solomon Feferman]] et al., eds. Oxford University Press.
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| * {{cite book| author=Hermann Grassmann| title=Lehrbuch der Arithmetik (A tutorial in arithmetic)| year=1861| publisher=Enslin| url=http://www.uni-potsdam.de/u/philosophie/grassmann/Werke/Hermann/Lehrbuch_der_Arithmetik_1861.pdf}}
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| * Hatcher, William S., 1982. ''The Logical Foundations of Mathematics''. Pergamon. Derives the Peano axioms (called '''S''') from several [[axiomatic set theories]] and from [[category theory]].
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| *[[David Hilbert]],1901, "Mathematische Probleme". ''Archiv der Mathematik und Physik'' 3(1): 44–63, 213–37. English translation by Maby Winton, 1902, "[http://aleph0.clarku.edu/~djoyce/hilbert/problems.html Mathematical Problems,]" ''Bulletin of the American Mathematical Society'' 8: 437–79.
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| * Kaye, Richard, 1991. ''Models of Peano arithmetic''. Oxford University Press. ISBN 0-19-853213-X.
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| <!-- * [[Charles Sanders Peirce|Peirce, C. S.]] (1881), "On the Logic of Number", ''American Journal of Mathematics'' v. 4, pp. [http://books.google.com/books?id=LQgPAAAAIAAJ&jtp=85 85-95]. Reprinted (CP 3.252-88), (W 4:299-309). -->
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| * {{cite news|last=Peirce|first=C. S.|authorlink=Charles Sanders Peirce|title=On the Logic of Number |url=http://books.google.com/books?id=LQgPAAAAIAAJ&jtp=85|journal=American Journal of Mathematics|volume=4|year=1881
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| |number=1–4|pages=85–95|doi=10.2307/2369151|mr=1507856 |jstor=2369151}} Reprinted (CP 3.252-88), (W 4:299-309).
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| * Paul Shields. (1997), "Peirce's Axiomatization of Arithmetic", in Houser ''et al.'', eds., ''Studies in the Logic of Charles S. Peirce''.
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| *[[Patrick Suppes]], 1972 (1960). ''Axiomatic Set Theory''. Dover. ISBN 0-486-61630-4. Derives the Peano axioms from [[ZFC]].
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| *[[Alfred Tarski]], and Givant, Steven, 1987. ''A Formalization of Set Theory without Variables''. AMS Colloquium Publications, vol. 41.
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| *[[Edmund Landau]], 1965 ''Grundlagen Der Analysis''. AMS Chelsea Publishing. Derives the basic number systems from the Peano axioms. English/German vocabulary included. ISBN 978-0-8284-0141-8
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| *{{cite book | author = [[Jean van Heijenoort]], ed.| coauthors = | title = From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931| edition = 3rd | publisher = Harvard University Press | location = Cambridge, Mass| year = 1967, 1976 3rd printing with corrections| isbn = 0-674-32449-8 (pbk.)}} Contains translations of the following two papers, with valuable commentary:
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| **[[Richard Dedekind]], 1890, "Letter to Keferstein." pp. 98–103. On p. 100, he restates and defends his axioms of 1888.
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| **[[Giuseppe Peano]], 1889. ''Arithmetices principia, nova methodo exposita'' (The principles of arithmetic, presented by a new method), pp. 83–97. An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations.
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| {{PlanetMath attribution|id=3301|title=PA}}
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| ==External links==
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| *[[Internet Encyclopedia of Philosophy]]: "[http://www.utm.edu/research/iep/p/poincare.htm Henri Poincare]" by Mauro Murzi. Includes a discussion of Poincaré's critique of the Peano's axioms.
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| * [http://www.ltn.lv/~podnieks/gt3.html First-order arithmetic], a chapter of a book on the [[incompleteness theorem]]s by Karl Podnieks.
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| * {{springer|title=Peano axioms|id=p/p071880}}
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| * {{planetmath reference|id=2789|title=Peano arithmetic}}
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| * {{MathWorld | urlname=PeanosAxioms | title=Peano's Axioms}}
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| * [http://www.math.uwaterloo.ca/~snburris/htdocs/scav/dedek/dedek.html What are numbers, and what is their meaning?: Dedekind] commentary on Dedekind's work, Stanley N. Burris, 2001.
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| {{DEFAULTSORT:Peano Axioms}}
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| [[Category:1889 introductions]]
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| [[Category:Mathematical axioms]]
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| [[Category:Formal theories of arithmetic]]
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| [[Category:Logic in computer science]]
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| [[Category:Mathematical logic]]
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| [[hu:Giuseppe Peano#A természetes számok Peano-axiómái]]
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