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| − | A system of five axioms for the set of natural numbers $\mathbb{N}$ and a function $S$ (successor) on it, introduced by G. Peano (1889):
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| − | # $0 \in \mathbb{N}$
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| − | # $x \in \mathbb{N} \to Sx \in \mathbb{n}$
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| − | # $x \in \mathbb{N} \to Sx \neq 0$
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| − | # $x \in \mathbb{N} \wedge y \in \mathbb{N} \wedge Sx =Sy \to x = y$
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| − | # $0 \in M \wedge \forall x (x\in M \to Sx\in M) \to \mathbb{N} \subseteq M$ for any property $M$ (axiom of induction).
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| − | In the first version of his system, Peano used $1$ instead of $0$ in axioms 1, 3, and 5. Similar axioms were proposed by R. Dedekind (1888).
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| − | In the Peano axioms presented above, the axiom of induction (axiom 5) is a statement in second-order language. Dedekind proved that all systems of Peano axioms with such a second-order axiom of induction are categorical. That is, any two systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188010.png" /> satisfying them are isomorphic. The isomorphism is determined by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188011.png" />, where
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188012.png" /></td> </tr></table>
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188013.png" /></td> </tr></table>
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| − | The existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188014.png" /> for all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188015.png" /> and the mutual single-valuedness for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188016.png" /> are proved by induction.
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| − | Peano's axioms make it possible to develop number theory and, in particular, to introduce the usual arithmetic functions and to establish their properties.
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| − | All the axioms are independent, but
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| − | and
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| − | can be combined to a single one:
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188017.png" /></td> </tr></table>
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| − | if one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188018.png" /> as
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188019.png" /></td> </tr></table>
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| − | The independence of Peano’s axioms is proved by exhibiting, for each axiom, a model for which the axiom considered is false, but for which all the other axioms are true.
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| − | For example:
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| − | * for axiom 1, such a model is the set of natural numbers beginning with $1$
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| − | * for axiom 2, it is the set $\mathbb{N} \cup \{1/2\}$, with $S0 = 1/2$ and $S1/2 =1$
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| − | * for axiom 3, it is the set $\{0\}$
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| − | * for axiom 4, it is the set $\{0, 1\}$, with $S0 = S1 = 1$
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| − | * for axiom 5, it is the set $\mathbb{N} \cup \{-1\}$
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| − | Using this method, Peano provided a proof of independence for his axioms (1891).
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| − | Sometimes one understands by the term ''Peano arithmetic'' the system in the first-order language
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| − | ::with the function symbols
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| − | ::::$S, +, \cdot$,
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| − | ::consisting of axioms
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| − | ::::$Sx\neq 0$ and $Sx = Sy \to x = y$
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| − | ::defining equalities for $+$ and $\cdot$
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| − | ::::$x + 0 = x$ and $x + Sx = S(x + y)$
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| − | ::::$x \cdot 0 = 0$ and $x \cdot S(y) = x \cdot y + x$
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| − | ::and with the induction scheme
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| − | ::::$A (0) \wedge \forall x (A(x) \to A(Sx)) \to \forall x A(x)$
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| − | where $A$ is an arbitrary formula, known as the induction formula (see [[Arithmetic, formal|Arithmetic, formal]]).
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| − | ====References====
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| − | * S.C. Kleene, ''Introduction to Metamathematics'', North-Holland (1951).
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| − | ====Comments====
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| − | The system of Peano arithmetic in first-order language, mentioned at the end of the article, is no longer categorical (cf. also [[Categoric system of axioms|Categoric system of axioms]]), and gives rise to so-called non-standard models of arithmetic.
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| − | ====References====
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| − | * H.C. Kennedy, ‘’Peano. Life and works of Giuseppe Peano’’, Reidel (1980).
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| − | * H.C. Kennedy, ‘’Selected works of Giuseppe Peano’’, Allen & Unwin (1973).
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| − | * E. Landau, ‘’Grundlagen der Analysis’’, Akad. Verlagsgesellschaft (1930).
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