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A construction of type: type in Martin-Löf's partial type theory with one universe

Published online by Cambridge University Press:  12 March 2014

Erik Palmgren
Erik Palmgren*
Affiliation:
Department of Mathematics, Uppsala University, 752 38 Uppsala, Sweden
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Extract

In this note we construct Martin-Löf's inconsistent type theory, Type: Type (Martin-Löf [1971]), inside partial type theory with one universe. Thus adding a fixed point operator to type theory with one predicative universe gives impredicativity.

We may describe the theory Type:Type as follows. It contains the rules for the product construction (II) of Martin-Löf [1984] except the η-rule and it contains the usual rules for definitional equality (=). Moreover it contains the following strongly impredicative universe

This theory is inconsistent (i.e. every set is inhabited), and this is seen by proving a variant of the Burali—Forti paradox—Girard's paradox—cf. Troelstra and van Dalen [1988]. Coquand [199?] has shown that by adding the well-order type and the strong dependent sum to the universe, the fixed point operator becomes definable. It is an open problem whether it is definable without the well-order type. The present result could be seen as a converse, namely by adding the fixed point operator to type theory with one universe, Type:Type becomes definable and, as is already known, so does the well-order type.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 3 , September 1991 , pp. 1012 - 1015
Copyright
Copyright © Association for Symbolic Logic 1991

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References

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