@ARTICLE{Coquand91theparadox, author = {Thierry Coquand}, title = {The paradox of trees in Type Theory}, journal = {BIT}, year = {1991}, volume = {32} }

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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...

...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...

...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...

...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...

...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...

...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...