Universe (mathematics and logic)

A specified domain of objects over which variables range or within which a mathematical structure is defined. Its choice determines which objects and statements a formal theory can describe.

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Domain of discourse: The collection of objects over which variables in a logical expression range. In first-order logic, this is the universe that fixes the possible values of variables.
First-order logic: A formal logic with variables, predicates, functions, and quantifiers over individual objects. Its quantifiers range over a specified domain rather than over arbitrary properties.
Set-theoretic universe: A universe of sets considered within a particular set theory or model of set theory. This is a foundational setting in which the domain itself consists of sets.
Group (mathematics): A set with an associative operation, an identity element, and an inverse for every element. A group structure takes its elements as the universe for its algebraic laws.
Variable (mathematics): A symbol that stands for an unspecified or changing mathematical object. A variable ranges over the universe selected for its expression or theory.
Structure (mathematical logic): A domain together with interpretations of the symbols in a formal language. Its domain supplies the universe in which formulas receive truth values.
Proper class: A collection too large to be a set in a given set theory, such as the collection of all sets. Some proposed universes exceed the size allowed for sets and must be treated as classes.
Real number: An element of the complete ordered field that extends the rational numbers. Real analysis commonly fixes the real numbers as its domain.
Mathematical structure: A set equipped with specified relations, operations, or distinguished elements. A structure's universe is its underlying set of objects.
Model theory: The study of the relationships between formal languages and their mathematical structures. It examines how different universes can satisfy the same theory.