Can You Add Power‐Sets to Martin‐Lof's Intuitionistic Set Theory?

Maietti, Maria Emilia; Valentini, Silvio

doi:10.1002/malq.19990450410

Cited by 25 publications

(14 citation statements)

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“…58]. It was proved for toposes by Diaconescu [3], for constructive set theory by Goodman and Myhill [4], and for some intensional type theories e. g. by Lacas and Werner [5] and Maietti [6,7]. We give a somewhat different proof in this paper.…”

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confidence: 87%

EM + Ext₋ + AC_int is equivalent to AC_ext

Carlström

2004

Mathematical Logic Qtrly

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It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional ('type-theoretical') axiom of choice ACint, which is provable in Martin-Löf's type theory, and a weak extensionality principle (Ext−), which is provable in Martin-Löf's extensional type theory. In particular, EM is equivalent to ACext in extensional type theory.The following is the principle AC int of intensional choice: if A, B are sets and R a relation such thatIt follows from AC ext that surjective functions have right inverses: If = B is an equivalence relation on B andsurjectivity is an instance of the premise needed to apply intensional choice (switch A and B). It says that there is a function g :This, however, does not mean that g is extensional, that is, that it preserves equivalence relations. If = A is an equivalence relation on A and = B is an equivalence relation on B, it might very well happen that f preserves them but g does not. If both A and B are the set of Cauchy sequences of rational numbers, for instance, and = A is pointwise equality, while = B is real number equality, then the identity function is clearly extensional from A/= A to B/= B , but not in the other direction. Thus it has a right inverse (itself) which is not extensional. In fact, we have no reason to expect that we should be able to construct an extensional right inverse in this case.The principle AC ext of extensional choice states that there is an extensional function with the property required. To be precise, it states that if R is an extensional relation (that is, it respects the equivalence relations) and (∀x : A)(∃y : B) R(x, y) is true, then there is an extensional function f : A −→ B such that (∀x : A) R(x, f (x)) is true. As the before-mentioned example with real numbers indicates, one cannot justify AC ext constructively. In fact, it implies the principle of excluded middle. 1) Thus there is, from a constructive point of view, a big difference in status between the intensional and extensional axioms of choice. However, they are equivalent in ZF and other theories with sufficiently strong axioms for quotient sets, which explains why we are used to hear the name 'axiom of choice', with no mention of the extensionality.At the beginning we remark the following well-known proposition: * e-mail: jesper@math.su.se, http://www.math.su.se/∼jesper 1) This was left as an exercise by Bishop [2, p. 58]. It was proved for toposes by Diaconescu [3], for constructive set theory by Goodman and Myhill [4], and for some intensional type theories e. g. by Lacas and Werner [5] and Maietti [6,7]. We give a somewhat different proof in this paper.

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confidence: 87%

EM + Ext₋ + AC_int is equivalent to AC_ext

Carlström

2004

Mathematical Logic Qtrly

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“…The above definition, taken literally, is definitely too restrictive if the notion of set is interpreted as in type theory, where for instance P(X) is never a set (see [9]). Therefore one has to give up the fact that L is a set, and require L to be a collection (or category; see [10]).…”

Section: Infinitary Terms and Relationsmentioning

confidence: 98%

Pretopologies and a uniform presentation of sup-lattices, quantales and frames

Battilotti

Sambin

2006

Annals of Pure and Applied Logic

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We introduce the notion of infinitary preorder and use it to obtain a predicative presentation of sup-lattices by generators and relations. The method is uniform in that it extends in a modular way to obtain a presentation of quantales, as “sup-lattices on monoids”, by using the notion of pretopology. Our presentation is then applied to frames, the link with Johnstone’s presentation of frames is spelled out, and his theorem on freely generated frames becomes a special case of our results on quantales. The main motivation of this paper is to contribute to the development of formal topology. That is why all our definitions and proofs can be expressed within an intuitionistic and predicative foundation, like constructive type theory

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“…✷ After the previous corollary one can wonder weather the equality condition that we proposed is consistent at all. The answer is positive if we assume to work within Martin-Löf's type theory instead that within an impredicative type system, since in this case a model can be found within Zermelo-Fraenkel set theory with the axiom of choice (see [8]). …”

Section: The Power-set Constructormentioning

confidence: 99%

Extensionality Versus Constructivity

Valentini

2002

MLQ - Math. Log. Quart.

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Can You Add Power‐Sets to Martin‐Lof's Intuitionistic Set Theory?

Cited by 25 publications

References 6 publications

EM + Ext₋ + AC_int is equivalent to AC_ext

EM + Ext₋ + AC_int is equivalent to AC_ext

Pretopologies and a uniform presentation of sup-lattices, quantales and frames

Extensionality Versus Constructivity

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Can You Add Power‐Sets to Martin‐Lof's Intuitionistic Set Theory?

Cited by 25 publications

References 6 publications

EM + Ext− + ACint is equivalent to ACext

EM + Ext− + ACint is equivalent to ACext

Pretopologies and a uniform presentation of sup-lattices, quantales and frames

Extensionality Versus Constructivity

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EM + Ext₋ + AC_int is equivalent to AC_ext

EM + Ext₋ + AC_int is equivalent to AC_ext