Britannica
Sources used in this answer
- Metalogic - Background and typical problems
- Metalogic - Satisfaction of a theory by a structure: finite and infinite models
- Metalogic - Characterizations of the first-order logic
- Metalogic - Nonelementary logic and future developments
- History of logic - Development of model theory
- Metalogic - Syntax and semantics
How does model theory use interpretations called models?
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Metalogic
David Hilbert.
In model theory, a model serves as an interpretation of a formal language. To understand how this works, consider the following process:
- Defining the Language: A formal language is established using a collection of symbols for constants, functions, and relations. A theory is then formed by selecting a specific set of sentences from that language, often designated as the "true" or "distinguished" elements.
- Creating a Structure: A model is a structure that provides a domain of objects and assigns specific meanings to the language's symbols. It maps constants to objects in the domain and predicate or function symbols to relations or functions within that domain.
- Satisfaction: A structure is considered a model of a theory if all the distinguished sentences of that theory are true within that structure. This relationship is known as satisfaction.
By using models, logicians can study the semantic properties of theories, distinguishing them from purely syntactic proof theory. This approach allows for the classification of models and the exploration of how different structures satisfy the same set of axioms.123