Proving the artihmetic property of a theory

Question

When can we say a threorom is arithmetic? Is it by following the first 9 axioms of Peano arithmetics? Or should it only allow us to express idioms of natural numbers?

For example, can we add a claim stating there exists a real number (e.g. 1.5) to the rest of Peano's axioms and it will remain arithmetic?

Answer 1

If we are considering first-order languages, arithmetic is a collection of languages.

We can the symbols $0$ and $S$ and, in addition, some or all of : $<, +, \times$.

With them, we can formulate several theoris, according to the set of axioms adopted :

Peano arithmetic : $\mathsf {PA}$

Robinson arithemetic : $\mathsf {Q}$

Presburger arithmetic.

With this kind of f-o languages we can try to express the axiom :

there exists a real number (e.g. $1.5$)

with something like : $\exists z \ [z \times S(S(0)) = S(S(S(0)))]$ or $\exists z \ [z + z = S(S(S(0)))]$.

Of course, adding it to the nine Peano axioms we get a new theory $\mathsf {PA}_{1.5}$ whose model (if any) will be no more $\mathbb N$.