I am aware that assigning the type of Type to be Type (rather than stratifying to a hierarchy of types) leads to an inconsistency. But does this inconsistency allow the construction of a well-typed term with no normal form, or does it actually allow a proof of False? Are these two questions equivalent?



What is the manner of inconsistency of Girard's paradox in Martin Lof type theory
Girard's paradox constructs a non-normalizing proof of False. You could read Hurken's "A simplification of Girard's paradox", or maybe Kevin Watkin's [formalization in Twelf](http://www.cs.cmu.edu/~kw/research/hurkens95tlca.elf).


In general, these questions are not equivalent, though they often coincide. A "reasonable" type theory will by inspection have no *normal* proofs of False, and so then normalization implies consistency. The inverse (non-normalization => proof of False) is much less obvious, and it is certainly possible to construct reasonable paraconsistent type theories, where non-termination is confined under a monad and does not result in a proof of False.


