constructive set theoryの例文

• Other researchers have also modeled parametric polymorphism within constructive set theories.
• Thus the axiom of choice is not generally available in constructive set theory.
• It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.
• Bounded quantifiers are important in Kripke-Platek set theory and constructive set theory, where only predicative grounds.
• These include the program of intuitionism founded by constructive set theories such as IZF and the study of topos theory.
• While the earliest results were for constructive theories of arithmetic, many results are also known for constructive set theories ( Rathjen 2005 ).
• Modern constructive set theory includes the axiom of infinity from ZFC ( or a revised version of this axiom ) and the set of natural numbers.
• Some constructive set theories include weaker forms of the axiom of choice, such as the axiom of dependent choice in Myhill's set theory.
• A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.
• In constructive set theory, Myhill is known for proposing an axiom system that avoids the axiom of choice and the law of the excluded middle, known as Intuitionistic Zermelo Fraenkel.
• He also developed a constructive set theory based on natural numbers, functions, and sets, rather than ( as in many other foundational theories ) basing it purely on sets.
• Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle in constructive set theory.
• Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle in constructive set theory.
• :: : : You awake the impression that all general set theories are naive and thus to be discarded and only the constructive set theories with its formalized axioms are not naive.
• In constructive set theory, where the law of excluded middle does not necessarily hold, one can consider the relation'subquotient of'as replacing the usual order relation ( s ) on cardinals.
• In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of excluded middle ( unlike in Martin-L鰂 type theory, where it does not ).
• In constructive set theories with only the predicative separation, the form of " P " will be restricted to sentences with bound quantifiers only, giving only a restricted form of the law of the excluded middle.
• However, in certain axiom systems for constructive set theory, the axiom of choice does imply the law of the excluded middle ( in the presence of other axioms ), as shown by the Diaconescu-Goodman-Myhill theorem.
• An example of a Brouwerian counterexample of this type is Diaconescu's theorem, which shows that the full axiom of choice is non-constructive in systems of constructive set theory, since the axiom of choice implies the law of excluded middle in such systems.
• Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not simply "'True "'or "'False " '.