KnowraType theoryLinked fromLinked fromThe 35 pages that link to Type theory, each with the reason it gives.All 35Broader topic 1Related 19Narrower topic 2Compared with 13Set theoryCompared with: It offers a foundational alternative to treating all mathematical objects as sets.Bertrand RussellRelated: Russell proposed a hierarchy of types to block paradoxes like his own.Zermelo–Fraenkel set theoryCompared with: It is a foundational alternative that avoids treating every object as a set.Intuitionistic logicRelated: Intuitionistic logic underlies the propositions-as-types interpretation used in type-theoretic foundations.Russell's paradoxRelated: It blocks paradoxical constructions by restricting which objects can be related.Category theoryCompared with: Type theory offers a distinct foundation that can also express categorical structures.Principia MathematicaRelated: The ramified hierarchy of types blocks self-reference behind logical paradoxes.Proof theoryRelated: Proof theory studies the derivations and normalization behavior of type systems.Mathematical logicCompared with: It offers an alternative foundational framework in which types can encode propositions and proofs.ParadoxRelated: Russell's paradox helped motivate restrictions on which objects can be treated as sets.Curry–Howard correspondenceNarrower topic: Curry–Howard is one of the central bridges between type theory and logic.Formal systemRelated: It can serve as a foundation where propositions correspond to types and proofs to terms.Ernst ZermeloCompared with: It offers a foundational alternative to Zermelo’s set-based axiomatic approach.Proof assistantRelated: Many assistants use type theory to make proof checking part of type checking.Constructive proofRelated: Its propositions-as-types perspective makes constructive proofs correspond to witness-bearing terms.Foundations of mathematicsRelated: It offers an alternative foundation in which proofs and programs can share structure.Constructive mathematicsRelated: Proofs-as-programs interpretations represent propositions and their evidence using types.Cumulative hierarchyCompared with: Its stratification avoids treating all mathematical objects as levels of one cumulative set universe.Second-order logicCompared with: Some type theories encode higher-order reasoning through typed functions and propositions.Personality traitCompared with: Trait models usually describe degrees, not sharply separated personality types.LogicismBroader topic: Russell and Whitehead used a type hierarchy to block the paradox that damaged Frege’s system.CategoryRelated: Some type theories encode categorical structures and offer alternative foundations for them.ElementCompared with: Type membership resembles set membership but belongs to a different formal framework.Heyting algebraRelated: The propositions-as-types correspondence links intuitionistic proof rules to typed constructions.Axiomatic systemCompared with: It can serve as a foundation for mathematics through typed constructions rather than set-theoretic axioms.Axiom of Power SetCompared with: Many type-theoretic foundations represent collections without adopting this set-existence axiom.Higher-order logicRelated: Typed formulations assign distinct types to objects, predicates, and functions.ConstructivismRelated: Constructive type theories make proofs and the objects they construct part of one formal framework.New FoundationsNarrower topic: Stratification imports a type-like constraint into an untyped set-theoretic language.Formal scienceRelated: It connects mathematical foundations with programming-language design and proof assistants.Katharine Cook BriggsRelated: Briggs’s work helped bring a Jungian version of this approach into a widely used assessment.Conjunction introductionRelated: Product types model conjunction, with introduction constructing a pair.Axiom of Empty SetCompared with: Some type theories treat empty types rather than assuming an empty set as an axiom.Barbara H. ParteeRelated: Semantic types organize the function-and-argument structure used in formal analyses.Universe (mathematics and logic)Compared with: Type theories can organize objects into levels instead of placing everything in one domain.
KnowraType theoryLinked fromLinked fromThe 35 pages that link to Type theory, each with the reason it gives.All 35Broader topic 1Related 19Narrower topic 2Compared with 13Set theoryCompared with: It offers a foundational alternative to treating all mathematical objects as sets.Bertrand RussellRelated: Russell proposed a hierarchy of types to block paradoxes like his own.Zermelo–Fraenkel set theoryCompared with: It is a foundational alternative that avoids treating every object as a set.Intuitionistic logicRelated: Intuitionistic logic underlies the propositions-as-types interpretation used in type-theoretic foundations.Russell's paradoxRelated: It blocks paradoxical constructions by restricting which objects can be related.Category theoryCompared with: Type theory offers a distinct foundation that can also express categorical structures.Principia MathematicaRelated: The ramified hierarchy of types blocks self-reference behind logical paradoxes.Proof theoryRelated: Proof theory studies the derivations and normalization behavior of type systems.Mathematical logicCompared with: It offers an alternative foundational framework in which types can encode propositions and proofs.ParadoxRelated: Russell's paradox helped motivate restrictions on which objects can be treated as sets.Curry–Howard correspondenceNarrower topic: Curry–Howard is one of the central bridges between type theory and logic.Formal systemRelated: It can serve as a foundation where propositions correspond to types and proofs to terms.Ernst ZermeloCompared with: It offers a foundational alternative to Zermelo’s set-based axiomatic approach.Proof assistantRelated: Many assistants use type theory to make proof checking part of type checking.Constructive proofRelated: Its propositions-as-types perspective makes constructive proofs correspond to witness-bearing terms.Foundations of mathematicsRelated: It offers an alternative foundation in which proofs and programs can share structure.Constructive mathematicsRelated: Proofs-as-programs interpretations represent propositions and their evidence using types.Cumulative hierarchyCompared with: Its stratification avoids treating all mathematical objects as levels of one cumulative set universe.Second-order logicCompared with: Some type theories encode higher-order reasoning through typed functions and propositions.Personality traitCompared with: Trait models usually describe degrees, not sharply separated personality types.LogicismBroader topic: Russell and Whitehead used a type hierarchy to block the paradox that damaged Frege’s system.CategoryRelated: Some type theories encode categorical structures and offer alternative foundations for them.ElementCompared with: Type membership resembles set membership but belongs to a different formal framework.Heyting algebraRelated: The propositions-as-types correspondence links intuitionistic proof rules to typed constructions.Axiomatic systemCompared with: It can serve as a foundation for mathematics through typed constructions rather than set-theoretic axioms.Axiom of Power SetCompared with: Many type-theoretic foundations represent collections without adopting this set-existence axiom.Higher-order logicRelated: Typed formulations assign distinct types to objects, predicates, and functions.ConstructivismRelated: Constructive type theories make proofs and the objects they construct part of one formal framework.New FoundationsNarrower topic: Stratification imports a type-like constraint into an untyped set-theoretic language.Formal scienceRelated: It connects mathematical foundations with programming-language design and proof assistants.Katharine Cook BriggsRelated: Briggs’s work helped bring a Jungian version of this approach into a widely used assessment.Conjunction introductionRelated: Product types model conjunction, with introduction constructing a pair.Axiom of Empty SetCompared with: Some type theories treat empty types rather than assuming an empty set as an axiom.Barbara H. ParteeRelated: Semantic types organize the function-and-argument structure used in formal analyses.Universe (mathematics and logic)Compared with: Type theories can organize objects into levels instead of placing everything in one domain.