Intended Interpretations
Many formal languages are associated with a particular interpretation that is used to motivate them. For example, the first-order signature for set theory includes only one binary relation, ∈, which is intended to represent set membership, and the domain of discourse in a first-order theory of the natural numbers is intended be the set of natural numbers. However, there are always other unintended interpretations in which either the domain of discourse or the non-logical symbols are not given their intended meanings.
In the context of Peano arithmetic, the intended interpretation is called the standard model of arithmetic. It consists of the natural numbers with their ordinary arithmetical operations. All models that are isomorphic to the one just given are also called standard; these models all satisfy the Peano axioms. There are also non-standard models of the Peano axioms, which contain elements not correlated with any natural number.
Read more about this topic: Interpretation (logic)
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