Languages and Structures

Mathematical Logic · Axiom Academy

LESSON Languages and Structures Formal foundations of model theory: languages, signatures, and mathematical structures Mathematical Logic • Unit 7 - Model Theory A first-order language provides the vocabulary for expressing mathematical statements. It consists of symbols that represent constants, functions, and relations. Definition: First-Order Language A first-order language consists of: Function symbols: (each with arity ) Relation symbols: (each with arity ) Together with logical symbols: The signature (or vocabulary ) of a language specifies which constants, functions, and relations are available, along with their arities. is a set of function symbols with arities is a set of relation symbols with arities Identity constant, binary operation, unary inverse Two constants, two binary operations, one unary An L-structure (or model ) gives meaning to the symbols in a language by providing a domain and interpretations for all symbols. Constant interpretations: For each , an element Function interpretations: For each , a function Relation interpretations: For each , a relation The structure specifies the domain and the interpretation of each symbol in the signature. Let's see concrete examples of structures for different mathematical theories. The integers with standard addition form a group Real numbers with standard ordering A finite graph with 4 vertices

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