# Definition:Language of Arithmetic

## Definition

A **language of arithmetic** is a signature for predicate logic consisting of:

- the binary operation symbols: $+$ and $\cdot$
- the unary function symbol: $s$
- the binary relation symbol: $<$
- the constant symbol: $0$

## Standard Interpretation

The standard interpretation for this language is the set of integers with:

- $+$: Integer addition
- $\cdot$: Integer multiplication
- $s$: the successor function
- $<$: the usual strict ordering on the integers
- $0$: interpreted as $0 \in \Z$

## Comment

This is just one of many different signatures that could claim to be the **language of arithmetic**.