What model do you get from PA without addition and multiplication?

Question

I have the feeling that this question is trivial, but I cannot figure the answer by myself nor from the stuff I have read. So the question is if addition (and multiplication) can be shown as a theorem of PA (without the arithmetical axioms but with successor and induction). If not they are independent and first order PA+¬A has a model (A=addition + multiplication axioms). What does a model of such system looks like?

Answer 1

I assume you mean a standard first-order PA setup.

So we have a language $L$ with constant symbol $0$, unary function symbol $S$, and binary function symbols $f$ and $g$ (their common names, when they are written in the middle, are $+$ and $\times$).

Make an $L$-structure by using the non-negative integers as the underlying set, and interpreting $0$ and $S$ in the usual way. We want to interpret $f$ and $g$ so that the usual axioms about addition and multiplication are false. It is awfully easy to make things false. For example, interpret $f(x,y)$ as $x^2+2y^2$, and $g(x,y)$ as $2x^2+y^2$. Almost anything will work.