# Definition:Minimal Arithmetic

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## Definition

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

\((\text M 1)\) | $:$ | \(\displaystyle \forall x:\) | \(\displaystyle \map s x \ne 0 \) | |||||

\((\text M 2)\) | $:$ | \(\displaystyle \forall x, y:\) | \(\displaystyle \map s x = \map s y \implies x = y \) | |||||

\((\text M 3)\) | $:$ | \(\displaystyle \forall x:\) | \(\displaystyle x + 0 = x \) | |||||

\((\text M 4)\) | $:$ | \(\displaystyle \forall x, y:\) | \(\displaystyle x + \map s y = \map s {x + y} \) | |||||

\((\text M 5)\) | $:$ | \(\displaystyle \forall x:\) | \(\displaystyle x \cdot 0 = 0 \) | |||||

\((\text M 6)\) | $:$ | \(\displaystyle \forall x, y:\) | \(\displaystyle x \cdot \map s y = \paren {x \cdot y} + x \) | |||||

\((\text M 7)\) | $:$ | \(\displaystyle \forall x:\) | \(\displaystyle \neg x < 0 \) | |||||

\((\text M 8)\) | $:$ | \(\displaystyle \forall x, y:\) | \(\displaystyle x < \map s y \iff \paren {x < y \lor x = y} \) | |||||

\((\text M 9)\) | $:$ | \(\displaystyle \forall x:\) | \(\displaystyle 0 < x \iff x \ne 0 \) | |||||

\((\text M 10)\) | $:$ | \(\displaystyle \forall x, y:\) | \(\displaystyle \map s x < y \iff \paren {x < y \land y \ne \map s x} \) |

## Note

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

Note in particular that this is a finite list.