## The paradox of trees in Type Theory (1991)

Venue: | BIT |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Coquand91theparadox,

author = {Thierry Coquand},

title = {The paradox of trees in Type Theory},

journal = {BIT},

year = {1991},

volume = {32}

}

### OpenURL

### Abstract

. We show how to represent a paradox similar to Russell's paradox in Type Theory with W -types and a type of all types, and how to use this in order to represent a fixed-point operator in such a theory. It is still open whether or not such a construction is possible without the W -type. Introduction. It is known that Martin-Lof's Type Theory with one universe is inconsistent if this universe contains a name of itself (cf. [5,6,7,8]). Though it is possible under this hypothesis to produce a paradox similar to the one of Russell if we have an extensional equality (i.e. the equality described in [1]; see for instance [7] for a description of this paradox), it is not known yet if such a paradox occurs with the more intensional equality of Type Theory (as described in [3] 1 ), if we assume only as type constructors the \Pi and the \Sigma type operators (see [5,6] for a discussion of this problem). This question can be precised by asking whether there exists a term of Type Theory with a...

### Citations

242 |
Interpre'tation fonctionnelle et e'limination des coupures de l'arithme'tique d'ordre supe'rieur
- Girard
- 1972
(Show Context)
Citation Context ...r or not such a construction is possible without the W -type. Introduction. It is known that Martin-Lof's Type Theory with one universe is inconsistent if this universe contains a name of itself (cf. =-=[5,6,7,8]-=-). Though it is possible under this hypothesis to produce a paradox similar to the one of Russell if we have an extensional equality (i.e. the equality described in [1]; see for instance [7] for a des... |

145 |
Intuitionistic type theory. Bibliopolis
- Martin-Löf
- 1984
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Citation Context ...ins a name of itself (cf. [5,6,7,8]). Though it is possible under this hypothesis to produce a paradox similar to the one of Russell if we have an extensional equality (i.e. the equality described in =-=[1]-=-; see for instance [7] for a description of this paradox), it is not known yet if such a paradox occurs with the more intensional equality of Type Theory (as described in [3] 1 ), if we assume only as... |

119 |
The type theoretic interpretation of constructive set theory
- Aczel
- 2001
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Citation Context ...of itself, then any type named by an element of U is inhabited. This derivation can be seen as a derivation of Russell paradox in Type Theory via the coding of sets as trees. Such a coding is used in =-=[4]-=-, but the equality and the membership relations are defined by induction over a W type. It is enough in order to derive Russell paradox to use as equality the intensional equality, and as membership r... |

1 | A construction of Type:Type in Martin-Lof's partial type theory with one universe - Palmgren |

1 |
Type" is not a type, Principles of Programming language
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Citation Context ...r or not such a construction is possible without the W -type. Introduction. It is known that Martin-Lof's Type Theory with one universe is inconsistent if this universe contains a name of itself (cf. =-=[5,6,7,8]-=-). Though it is possible under this hypothesis to produce a paradox similar to the one of Russell if we have an extensional equality (i.e. the equality described in [1]; see for instance [7] for a des... |

1 |
The Computational Behaviour of
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Citation Context ...r or not such a construction is possible without the W -type. Introduction. It is known that Martin-Lof's Type Theory with one universe is inconsistent if this universe contains a name of itself (cf. =-=[5,6,7,8]-=-). Though it is possible under this hypothesis to produce a paradox similar to the one of Russell if we have an extensional equality (i.e. the equality described in [1]; see for instance [7] for a des... |