On inductive and Peano models


Mirko Stojaković




A Peano model $M$ is any triplet $(N,o,S)$ where $N$ is a set (of ``natural numbers''), $о$ is the element of $N$ and $S$ is a function from $N$ to $N$ for which the following conditions are valid: \qquad$A_1$: $Sx\neq о$ for every $x$ in $N$. \qquad$A_2$: For all $x$, $у$ in $N$ if $x\neq у$ then $Sx\neq y$. \qquad$A_3$: If $G$ is any subset of $N$ and from $0\in G$ and $x\in G$ it follows that $Sx\in G$, then $G=N$. An inductive model is any triplet $M$ in which only $A_3$ is guaranteed. Let $S_1,S_2$ be the respective binary operations $x+y$, $x.y$ over $N$, that is $S_1(x,y)=x+y$. $S_2(x,y)=xy$; then from the defining relations \qquad$S_1(x,0)=x$ (for all $x$ in $N$) \qquad$S_1(x,Sy)=S_0(x,S_1(x,y)$ (for all $x,у$ in $N$ where $S_0(u,v)=Sv)$ \qquad$S_2(x,0)=0$ \qquad$S_2(x,Sy)=S_1(x,S_2(x,y))$, we define the binary operation $S_3(x,y)=x^y$ by means of the relations \qquad$S_3(x,0)=1$, \qquad$S_3(x,Sy)=S_2(x,S3(x,y))$ which obviously generalizes by analogy to \qquad$S_n(x,0)=1$ $S_n(x,Sy)-S_{n-1}(x,S_n(x,y))$ We claim now that the triplet $M_n(a,q)=(G,a,T)$, $n=1, 2, 3,\ldots$ is a Peano model where $a\in G$; $a\in N$, $a\neq0,1$; $q\in N$, $q\neq0,1$ and for all $x$ in $N$ we have $Tx=S_n(x,q)$, the set $G$ being $\{a,S_n(a,q),\ S_n(a,S_n(a,q),\ldots\}$. Some relations between Peano models and inductive models are described and some features of the arithmetics in these general models are discussed.