[Agda] Re: eta expansion
Martin Escardo
m.escardo at cs.bham.ac.uk
Wed Mar 20 15:30:45 CET 2013
On 20/03/13 13:27, Altenkirch Thorsten wrote:
> The universes in Agda are open hence they don't admit an induction
> principle unlike the closed universes defined by induction-recursion. As
> a consequence all you can do a with type in a universe is extensional
> you cannot investigate how are the built. This is reflected by the
> univalence principle.
I agree with the view that universes are open ended. We know that they
are closed under certain constructions, but I don't want to say (or
think) that they are the least thing closed under those operations. Of
course, in the models, if you care about them, the universes will be
less open ended, and in some cases they can be defined as the
least-thing. But although I like models for several reasons, I also like
to take type theory as a foundational system (competing with set
theory), where my model is just a mental, intuitive one, and in this
view my universe may grow if I please, and more than that, it may have
things I am not aware of. Moreover, like all types, universes may have
plenty of non-syntactically-definable stuff. A weakening of univalence
is that any two isomorphic types have the same properties (or, put in
another way, any property of types is extensional, in that it has the
same value for any two isomorphic types). Univalence is the proper way
of stating that.
Of course, I am not forbiding any one of working with inductively
defined universes (and codes for the elements of the universe), and in
fact this can be done in Agda, and for some purposes I may wish to use
such inductively defined universes (maybe as HIT's as propoposed
earlier). But this is another universe, different from Set0, Set1, Set2,
... (the terminology Set for universes is probably not very good,
because its elements may not be hsets, but this is what Agda syntax has).
Martin
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