Cointerpretability

Cointerpretability is a concept in logic and mathematics, particularly in the field of model theory. It refers to the ability of two structures to interpret each other. This concept is fundamental in understanding the relationship between different mathematical structures and their logical interpretations.

Definition

In the context of model theory, a structure A is said to be cointerpretable with a structure B if there exists an interpretation of A in B and an interpretation of B in A. This means that the properties and relations of A can be expressed in terms of B, and vice versa.

Applications

Cointerpretability has wide-ranging applications in various fields of mathematics, including algebra, geometry, and number theory. It is used to establish connections between seemingly unrelated mathematical structures, thereby providing new insights and perspectives.

In logic, cointerpretability plays a crucial role in the study of equivalence relations on the class of all structures of a given language. It is also used in the investigation of definability and computability properties of structures.

Examples

One of the most famous examples of cointerpretability is the Lefschetz principle, which states that any statement in the language of rings is true for complex numbers if and only if it is true for all algebraically closed fields of characteristic zero.

Another example is the cointerpretability of the field of real numbers with the field of complex numbers, which is a fundamental result in model theory.

See also

• Interpretability
• Model theory
• Logic
• Mathematics

References

• Hodges, Wilfrid. Model Theory. Cambridge University Press, 1993.
• Marker, David. Model Theory: An Introduction. Springer, 2002.