I'm looking for a model of PA without induction whose order relation total and Anti-symmetric.
to be specific, satisfiying:
$\forall x \ (0 \neq S ( x ))$
$\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$
$\forall x \ (x + 0 = x )$
$\forall x, y \ (x + S( y ) = S( x + y ))$
I'll define the order as such:
$\forall x,y(x\ge y \iff \exists z(y+z=x))$
and I'll call it Antisymmetric if $\forall x,y(x\ge y \land x\le y\implies x=y)$
and total if $\forall x,y (x\ge y\lor x\le y)$
The standard non-induction model is one containing $a,b$ such that $a+1=b,b+1=a$ which contradicts antisymmetry.
There is a canonical example of a nonstandard model of "arithmetic without induction" - the set $\mathfrak{P}$ of polynomials in a single variable with integer coefficients which (are zero or) have positive leading term. The ordering is given by setting $f\trianglelefteq g$ iff $\lim_{x\rightarrow +\infty}g(x)-f(x)\ge 0$, and this is easily checked to be antisymmetric and total. (It is maybe a bit more natural to think of $\mathfrak{P}$ as the set of nonnegative elements of the ordered ring $\mathbb{Z}[x]$, with the ordering being as above.)
The structure $\mathfrak{P}$ has a number of nice properties: it has well-defined multiplication as well as addition, successor, and ordering, and it shows that induction is necessary to prove e.g. "Every number is either even or odd." Moreover, it's "computably describable" in a precise sense which we can't get once induction is demanded as well.