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Categorical semantics Categorical logic introduces the notion of structure valued in a category C with the classical model theoretic notion of a structure appearing in the particular case where C is the category of sets and functions. This notion has proven useful when the set-theoretic notion of a model lacks generality and/or is inconvenient.
A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley ( 1965 ) stating that if a first-order theory in a countable language is categorical in some uncountable cardinality , then it is categorical in all uncountable ...
a categorical model of bunched logic is a single category possessing two closed structures, one symmetric monoidal closed the other cartesian closed. A host of categorial models can be given using Day's tensor product construction. [8] Additionally, the implicational fragment of bunched logic has been given a game semantics. [9]
The field of formal semantics encompasses all of the following: The definition of semantic models; The relations between different semantic models; The relations between different approaches to meaning; The relation between computation and the underlying mathematical structures from fields such as logic, set theory, model theory, category ...
In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order ...
The aim of AST is to provide a uniform categorical semantics or description of set theories of different kinds (classical or constructive, bounded, predicative or impredicative, well-founded or non-well-founded, ...), the various constructions of the cumulative hierarchy of sets, forcing models, sheaf models and realisability models. Instead of ...
For this reason, it is used throughout mathematics. Applications to mathematical logic and semantics (categorical abstract machine) came later. Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundation of mathematics. A topos can also be considered as a specific type of category ...
A theory is called categorical if it has only one model, up to isomorphism. This term was introduced by Veblen (1904) , and for some time thereafter mathematicians hoped they could put mathematics on a solid foundation by describing a categorical first-order theory of some version of set theory.