# Definition:Minimal Arithmetic

## Definition

Minimal arithmetic is the set $Q$ of theorems of the recursive set of sentences in the language of arithmetic containing exactly:

 $(\text M 1)$ $:$ $\ds \forall x:$ $\ds \map s x \ne 0$ $(\text M 2)$ $:$ $\ds \forall x, y:$ $\ds \map s x = \map s y \implies x = y$ $(\text M 3)$ $:$ $\ds \forall x:$ $\ds x + 0 = x$ $(\text M 4)$ $:$ $\ds \forall x, y:$ $\ds x + \map s y = \map s {x + y}$ $(\text M 5)$ $:$ $\ds \forall x:$ $\ds x \cdot 0 = 0$ $(\text M 6)$ $:$ $\ds \forall x, y:$ $\ds x \cdot \map s y = \paren {x \cdot y} + x$ $(\text M 7)$ $:$ $\ds \forall x:$ $\ds \neg x < 0$ $(\text M 8)$ $:$ $\ds \forall x, y:$ $\ds x < \map s y \iff \paren {x < y \lor x = y}$ $(\text M 9)$ $:$ $\ds \forall x:$ $\ds 0 < x \iff x \ne 0$ $(\text M 10)$ $:$ $\ds \forall x, y:$ $\ds \map s x < y \iff \paren {x < y \land y \ne \map s x}$

## Note

These are just the usual axioms of arithmetic, except for the inductive axioms.

Note in particular that this is a finite list.