INRIA, and University of Goteborg/Chalmers; Abstract.
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. 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 ; see for instance  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  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...