Definition:Language of Arithmetic

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Definition

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

• the binary operation symbols: $+$ and $\times$
• 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
• $\times$: 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.

Sources

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