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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):
+
Each statement of a syllogism is one of 4 types, as follows:
  
# $0 \in \mathbb{N}$
+
::{| 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''' $B$ ||
 
+
|-
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
+
| 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>
 
 
 
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 and, 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 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.
 
For example:
 
* for axiom 1, such a model is the set 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 $\{0\}$
 
* for axiom 4, it is the set $\{0, 1\}$, with $S0 = S1 = 1$
 
* for axiom 5, it is the set $\mathbb{N} \cup \{-1\}$
 
 
 
Using this method, Peano provided a proof of independence for his axioms (1891).
 
 
 
Sometimes one understands by the term ''Peano arithmetic'' the system in the first-order language
 
 
 
::with the function symbols
 
::::$S, +, *$,
 
::consisting of axioms
 
::::$Sx\neq 0$ and $Sx = Sy \to x = y$
 
::defining equalities for
 
::::$+$ and $*$
 
::and with the induction scheme
 
::::$A (0) \wedge \forall x (A(x) \to A(Sx)) \to \forall x A(x)$
 
<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====
 
 
 
* S.C. Kleene, ''Introduction to Metamathematics'', North-Holland (1951).
 
 
 
====Comments====
 
 
 
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.
 
 
 
====References====
 
 
 
* H.C. Kennedy, ‘’Peano. Life and works of Giuseppe Peano’’, Reidel (1980).
 
* H.C. Kennedy, ‘’Selected works of Giuseppe Peano’’, Allen & Unwin (1973).
 
* E. Landau, ‘’Grundlagen der Analysis’’, Akad. Verlagsgesellschaft (1930).
 

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=36476