Interpretation (model Theory) - Definition

Definition

An interpretation of M in N with parameters (or without parameters, respectively) is a pair where n is a natural number and is a surjective map from a subset of Nn onto M such that the -preimage (more precisely the -preimage) of every set X ⊆ Mk definable in M by a first-order formula without parameters is definable (in N) by a first-order formula with parameters (or without parameters, respectively). Since the value of n for an interpretation is often clear from context, the map itself is also called an interpretation.

To verify that the preimage of every definable (without parameters) set in N is definable in M (with or without parameters), it is sufficient to check the preimages of the following definable sets:

• the domain of N;
• the diagonal of N;
• every relation in the signature of N;
• the graph of every function in the signature of N.

In model theory the term definable often refers to definability with parameters; if this convention is used, definability without parameters is expressed by the term 0-definable. Similarly, an interpretation with parameters may be referred to as simply an interpretation, and an interpretation without parameters as a 0-interpretation.