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| − | A system of five axioms for the set of natural numbers $\mathbb{N}$ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718801.png" /> and a function $S$ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718802.png" /> (successor) on it, introduced by G. Peano (1889):
| + | Each statement of a syllogism is one of 4 types, as follows: |
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| − | # $0 \in \mathbb{N}$
| + | ::{| class="wikitable" |
| − | # $x \in \mathbb{N} \to Sx \in \mathbb{n}$
| + | |- |
| − | # $x \in \mathbb{N} \to Sx \neq 0$
| + | ! Type !! Statement !! Alternative |
| − | # $x \in \mathbb{N} \wedge y \in \mathbb{N} \wedge Sx =Sy \to x = y$
| + | |- |
| − | # $0 \in M \wedge \forall x (x\in M \to Sx\in M) \to \mathbb{N} \subseteq M$ for any property $M$ (axiom of induction).
| + | | style="text-align: center;" | '''A''' || '''All''' $A$ '''are''' $B$ || |
| − | | + | |- |
| − | 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).
| + | | style="text-align: center;" | '''I''' || '''Some''' $A$ '''are''' $B$ || |
| − | | + | |- |
| − | Peano's axioms 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
| + | | style="text-align: center;" | '''E''' || '''No''' $A$ '''are''' $B$ || (= '''All''' $A$ '''are not''' $B$) |
| − | | + | |- |
| − | <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>
| + | | style="text-align: center;" | '''O''' || '''Not All''' $A$ '''are''' $B$ || (= '''Some''' $A$ '''are not''' $B$) |
| − | | + | |} |
| − | <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 the axioms is proved by exhibiting, for each axiom, a model for which the other axioms are true, but the one being considered is false:
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| − | * For Axiom 1, such a model is the series 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 $\{0\}$, with $S0 = 0$
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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 $\mathbb{N} \cup \{-1\}$, with $S-1 = -1$
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| − | Sometimes one understands by Peano arithmetic the system in the first-order language with the function symbols $S, +, *$<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188027.png" />, consisting of the axioms
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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/p07188028.png" /></td> </tr></table>
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| − | defining equalities for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188030.png" />, and the induction scheme
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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/p07188031.png" /></td> </tr></table>
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188032.png" /> is an arbitrary formula, known as the induction formula (see [[Arithmetic, formal|Arithmetic, formal]]).
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| − | | |
| − | ====References==== | |
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)</TD></TR></table>
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| − | ====Comments====
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| − | The system of Peano arithmetic mentioned at the end of the article above 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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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.C. Kennedy, "Peano. Life and works of Giuseppe Peano" , Reidel (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H.C. Kennedy, "Selected works of Giuseppe Peano" , Allen & Unwin (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Landau, "Grundlagen der Analysis" , Akad. Verlagsgesellschaft (1930)</TD></TR></table>
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