A More General Type Theoretic Interpretation Of Constructive Set Theory
نویسندگان
چکیده
منابع مشابه
The generalised type-theoretic interpretation of constructive set theory
We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in ty...
متن کاملCollection Principles in Dependent Type Theory
We introduce logic-enriched intuitionistic type theories, that extend intuitionistic dependent type theories with primitive judgements to express logic. By adding type theoretic rules that correspond to the collection axiom schemes of the constructive set theory CZF we obtain a generalisation of the type theoretic interpretation of CZF. Suitable logic-enriched type theories allow also the study...
متن کاملCharacterizing the interpretation of set theory in Martin-Löf typetheory
Constructive Zermelo-Fraenkel set theory, CZF, can be interpreted in Martin-Löf type theory via the so-called propositions-as-types interpretation. However, this interpretation validates more than what is provable in CZF. We now ask ourselves: is there a reasonably simple axiomatization (by a few axiom schemata say) of the set-theoretic formulae validated in Martin-Löf type theory? The answer i...
متن کاملHeyting-valued interpretations for Constructive Set Theory
The theory of locales [23] has a twofold interplay with intuitionistic mathematics: first of all, the internal logic of toposes and intuitionistic set theories provide suitable settings for the development of the theory of locales [24], and secondly, the notion of a locale determines two important forms of toposes and of interpretations for intuitionistic set theories, namely localic toposes [2...
متن کاملRealizability for Constructive Zermelo-Fraenkel Set Theory
Constructive Zermelo-Fraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a type-theoretic model. Aczel showed that it has a formulae-as-types interpretation in Martin-Löf’s intuitionist theory of types [14, 15]. This paper, thoug...
متن کامل