Re: [HoTT] Impredicative set + function extensionality + proof irrelevance consistent?

[Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>]

Hi Thomas,

we have already established that my argument was incorrect (for the reasons you state) and I was misinformed about the behaviour of Lean. 



>
>Another gap in Thorsten's argument is the following. Though Single(a) and
>Single(a') are isomorphic in order to conclude that they are propositionally
>equal they would have to be elements of a univalent universe BUT I don't see
>where such a universe should come from in the general topos case!

I don’t understand this point. In a type theoretic implementation of a topos Single(a) and Single(a’) would be propositionally equal due to propositional extensionality. The only additional assumption I need to make is that the universe of proposition is strict, e.g. we have that El(A -> B) is definitionally equal to  EL(A) -> El(B). This seems to be quite natural from the point of type theory where universes are usually strict and moreover this is true in any univalent category giving rise to a topos. 

Thorsten




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