Definition:Language of Arithmetic

Definition
The language of arithmetic is the first-order signature consisting of:
 * the binary function symbols $+$, $\cdot$
 * the unary function symbol $s$
 * the binary relation symbol $<$
 * the constant symbol $0$

Note
The standard interpretation for this language is the set of integers with:
 * $+$ addition
 * $\cdot$ multiplication
 * $s$ the successor function
 * $<$ the usual strict ordering on the integers
 * $0$ interpreted as $0\in \mathbb{Z}$

There are many different signatures that could claim to be the language of arithmetic, this is just one. For example, we could add constant symbols for each integer if we constrain their interpretation using axioms, since all of these integers can be referenced in a formula anyway using $0$ and an adequate number of applications of $s$.