Modellelméleti

Modellelméleti is a Hungarian term that translates to "model theory" in English. Model theory is a branch of mathematical logic that studies the relationship between formal languages and their interpretations, called models. It investigates how mathematical structures can be described using formal languages and how the properties of these structures can be deduced from the logical formulas that describe them.

At its core, model theory deals with the semantics of formal systems. A formal language consists of a set of symbols and rules for forming well-written formulas. A model is an interpretation of this language that assigns meaning to the symbols and formulas. This typically involves defining a domain of objects and interpreting the non-logical symbols as relations and functions on that domain. The central question in model theory is the relationship between the syntax of a formal language and the properties of its models.

Key concepts in model theory include elementary equivalence, embeddings, and isomorphisms. Two models are elementarily equivalent if they satisfy the same set of first-order sentences. An elementary embedding preserves this property for substructures. Model theory has applications in various fields of mathematics, including set theory, algebra, and topology, as well as in computer science, particularly in areas like database theory and automated reasoning. It provides a powerful framework for understanding the expressive power of formal languages and the nature of mathematical structures.