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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):
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Each statement of a syllogism is one of 4 types, as follows:
  
# $0 \in \mathbb{N}$ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718803.png" />
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::{| class="wikitable"
# $x \in \mathbb{N} \to Sx \in \mathbb{n}$ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718804.png" />
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|-
# $x \in \mathbb{N} \to Sx \neq 0$ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718805.png" />
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! Type !! Statement !! Alternative
# $x \in \mathbb{N} \wedge y \in \mathbb{N} \wedge Sx =Sy \to x = y$  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718806.png" />
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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 $M$ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718807.png" /> for any property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718808.png" /> (axiom of induction).
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| style="text-align: center;" | '''A''' || '''All''' $A$ '''are''' $B$ ||
 
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|-
In the first version, Peano used $1$ instead of $0$ in Axioms 1, 3, and 5. Similar axioms were proposed by R. Dedekind (1888).
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| style="text-align: center;" | '''I''' || '''Some''' $A$ '''are''' $B$ ||
 
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|-
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
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| style="text-align: center;" | '''E''' || '''No''' $A$ '''are''' $B$ || (= '''All''' $A$ '''are not''' $B$)
 
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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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| style="text-align: center;" | '''O''' || '''Not All''' $A$ '''are''' $B$ || (= '''Some''' $A$ '''are not''' $B$)
 
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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>
 
 
 
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. Peano's axioms make it possible to develop number theory; in particular, to introduce the usual arithmetic functions and to establish their properties. All the axioms are independent, but
 
 
 
and
 
 
 
can be combined to a single one:
 
 
 
<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>
 
 
 
if one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188018.png" /> as
 
 
 
<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>
 
 
 
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:
 
* For Axiom 1, such a model is the series of natural numbers beginning with $1$
 
* For Axiom 2, it is the set $\mathbb{N} \cup \{1/2\}$, with $S0 = 1/2 and S1/2 =1$
 
* For Axiom 3, it is the set $\{0\}$, with $S0 = 0$
 
* For Axiom 4, it is the set $\{0, 1\}$, with $S0 = S1 = 1$
 
* For Axiom 5, it is the set $\mathbb{N} \cup \{-1\}$, with $S-1 = -1$
 
 
 
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
 
 
 
<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>
 
 
 
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
 
 
 
<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>
 
 
 
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]]).
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1951)</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
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.
 
 
 
====References====
 
<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 &amp; Unwin  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Landau,  "Grundlagen der Analysis" , Akad. Verlagsgesellschaft  (1930)</TD></TR></table>
 

Latest revision as of 13:39, 14 October 2015

Each statement of a syllogism is one of 4 types, as follows:

Type Statement Alternative
A All $A$ are $B$
I Some $A$ are $B$
E No $A$ are $B$ (= All $A$ are not $B$)
O Not All $A$ are $B$ (= Some $A$ are not $B$)
How to Cite This Entry:
Whayes43. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whayes43&oldid=36467