## Extensionality versus constructivity (2000)

Venue: | Mathematical logic Quarterly |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Valentini00extensionalityversus,

author = {Silvio Valentini},

title = {Extensionality versus constructivity},

journal = {Mathematical logic Quarterly},

year = {2000},

volume = {42},

pages = {179--187}

}

### OpenURL

### Abstract

We will analyze some extensions of Martin-Löf’s constructive type theory by means of extensional set constructors and we will show that often the most natural requirements over them lead to classical logic or even to inconsistency. 1

### Citations

82 |
Lambda Calculi with Types, in Handbook of Logic
- Barendregt
- 1992
(Show Context)
Citation Context ...e exists a type B and two elements code ∈ prop → B and decode ∈ B → prop such that (∀A ∈ prop) A ↔ decode(code(A)) then we obtain an inconsistent theory since the inconsistent lambda-calculus λU (see =-=[Bar92]-=- for its definition) can be embedded into it. Now, note that the possibility to build the power-set of the one-element set, together with the required logical equivalence condition for the membership ... |

65 |
Metamathematical investigations on a calculus of constructions
- Coquand
- 1990
(Show Context)
Citation Context ...that also this kind of extensionality is dangerous if we are working within a framework which allows to quantify over the collection of all the propositions. In fact in this case we can argue like in =-=[Coq90]-=- (or also [Jac89]) where it is proved that if there exists a type B and two elements code ∈ prop → B and decode ∈ B → prop such that (∀A ∈ prop) A ↔ decode(code(A)) then we obtain an inconsistent theo... |

34 | Inductively generated formal topologies
- Coquand, Sambin, et al.
(Show Context)
Citation Context ...tin-Löf’s type theory appears immediately as soon as one tries to develop within such a framework a piece of actual mathematics, as we did with constructive topology (see the introductory sections in =-=[CSSV]-=-). For instance, one would like to be able to work with power-sets, in order to quantify on the elements of the collection of the subsets of a set, or with quotient-sets, in order to be able to constr... |

26 |
Axiom of choice and complementation
- Diaconescu
- 1975
(Show Context)
Citation Context ... provable within Martin-Löf’s type theory, implies 5sclassical logic. What we will formalize here within Martin-Löf’s type theory is one of the many re-writing of the original result by Diaconescu in =-=[Dia75]-=- that we found in [Bel88] (in order to see a fully detailed proof of this formalization inside Martin-Löf’s intensional type theory with a power-set constructor see [MV99]). First note that for any U,... |

21 |
Toposes & Local Set Theories: an Introduction
- Bell
- 1988
(Show Context)
Citation Context ...öf’s type theory, implies 5sclassical logic. What we will formalize here within Martin-Löf’s type theory is one of the many re-writing of the original result by Diaconescu in [Dia75] that we found in =-=[Bel88]-=- (in order to see a fully detailed proof of this formalization inside Martin-Löf’s intensional type theory with a power-set constructor see [MV99]). First note that for any U, V ∈ P(⊤), if decode(U) ∨... |

11 |
About effective quotients in Constructive Type Theory
- Maietti
(Show Context)
Citation Context ... quotient set V obtained by using the first universe U0, which contains the codes of all the small sets, and the equivalence relation of equi-provability between small types (for a detailed proof see =-=[Mai99]-=-). 5 The extensional two-subset set In the previous sections we proved that by adding extensional set-constructors to Martin-Löf’s type theory one can obtain classical logic. The obvious question is: ... |

2 |
The inconsistency of higher order extensions of Martin-Löf’s type theory
- Jacobs
- 1989
(Show Context)
Citation Context ...nd of extensionality is dangerous if we are working within a framework which allows to quantify over the collection of all the propositions. In fact in this case we can argue like in [Coq90] (or also =-=[Jac89]-=-) where it is proved that if there exists a type B and two elements code ∈ prop → B and decode ∈ B → prop such that (∀A ∈ prop) A ↔ decode(code(A)) then we obtain an inconsistent theory since the inco... |