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.