## Franklin's Notes

### Weaker theories of arithmetic

In first-order logic , the theory of Peano Arithmetic is the canonical axiomatization of arithmetic and number theory. However, there are a couple other noteworthy formal systems that are weaker than Peano Arithmetic.

• Robinson Arithmetic, often denoted $\mathrm{Q}$, is obtained by replacing the axiom schema of induction $(7_\varphi)$ by a single axiom $(8)$, which is as follows: Intuitively, this states that each nonzero number is the successor of some other number. This theory is incomplete.

• Presburger Arithmetic, or additive number theory, consists of the reduced language $\mathscr{L}={+,S,0}$ and axioms $(1)-(4),(7_\varphi)$ of Peano Arithmetic . This theory is complete but not finitely axiomatizable.

• The even weaker system consisting of the language $\mathscr{L}={S,0}$ and the axioms $(1),(2),$ and $(7_\varphi)$, is complete, but still not finitely axiomatizable, despite the fact that it is much less expressie than additive number theory.