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A system of five axioms for the set of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718801.png" /> and a function <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:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718803.png" />;
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::{| class="wikitable"
 
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|-
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718804.png" />;
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! Type !! Statement !! Alternative
 
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|-
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718805.png" />;
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| style="text-align: center;" | '''A''' || '''All''' $A$ '''are''' $B$ ||
 
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|-
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p0718806.png" />;
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| style="text-align: center;" | '''I''' || '''Some''' $A$ '''are''' $B$ ||
 
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|-
5) <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;" | '''E''' || '''No''' $A$ '''are''' $B$ || (= '''All''' $A$ '''are not''' $B$)
 
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|-
In the first version 1 was used instead of 0. Similar axioms were proposed by R. Dedekind (1888). 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;" | '''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/p07188012.png" /></td> </tr></table>
 
 
 
<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 is proved by exhibiting a model on which all the axioms are true except one. For
 
 
 
such a model is the series of natural numbers beginning with 1; for
 
 
 
it is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188022.png" />; for
 
 
 
the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188023.png" />; for
 
 
 
the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188024.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188025.png" />; for
 
 
 
the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071880/p07188026.png" />.
 
 
 
Sometimes one understands by Peano arithmetic the system in the first-order language with the function symbols <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=36464