[Agda] Agda with excluded middle is inconsistent

[Thorsten Altenkirch]

Hi Noam,

>> Surely, type constructors should not be injective in general. A definition
>> of the form
>> data I : (Set -> Set) -> Set where
>> should be expandable by an annonymous declaration
>> I : (Set -> Set) -> Set
>> I F = data {}
>> in an analogous way a named function definition can be expanded by
>> definition and a lambda abstraction
> 
> ...I'm not sure about this expansion, and the analogy.  I think the
> question is whether we view the declaration of type constructors in
> the same way that we view the declaration of term constructors (i.e.,
> as stipulating a new way of constructing things, in this case
> inhabitants of Set).  If we do, then type constructors are by
> definition injective.

I don't see the analogy. A term constructor is injective because an inductive type is freely generated from constructors. Indeed, we can show that the structure map of an initial algebra is an isomorphism and hence injective. However the universe of sets is not freely constructed from type constructors but they are really definitions.

We should justify principles like this (i.e. all type constructors are injective) by a semantic explanation or by a translation into a well understood core language e.g. (extensional) Marttin-Loef Type Theory with W-types and universes. I don't know any such explanation which would justify this principle. Moreover, Chung's example shows that this is not possible (without giving up something else) since pure type theory is consistent with EM.

Cheers,
Thorsten