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

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

On 17/12/2017, 10:21, "Thomas Streicher"
<stre...@mathematik.tu-darmstadt.de> wrote:

>> >Moreover, toposes will not validate univalence. As pointed out to me
>> >by Martin Escardo you can't even formulate it because in arbitrary
>> >toposes you don't have universe (not to speak of univalent ones).
>> 
>> I was only talking about the special case of (-1)-univalence or
>> propositional extensionality that is for P,Q : Prop
>> 
>> (P <-> Q) -> (P = Q)
>
>This would be ok but you assume also that Single(True) and
>Single(false) are propositions but they are just both ISOMORPHIC to 1
>in Omega = Prop. That's where you apply UA for (sub)singletons
>UNCONSCIOUSLY.

I wasn't defending the inconsistency derivation I posted earlier, as Mike
pointed out it, it ignores type annotations.

However, I don't understand your point. Surely UA for subsingletons is
exactly propositional extensionality.

>The inconsistency proof you have given is a slightly distorted version
>of the following ridiculous argument: {a} and {b} are isomorphic thus
>equal for which reason a = b.

Indeed, but we are not in set theory. In set level HoTT Single(True) =
Single(False) but True != False.

Whenever we can prove that a type is A propositional, that is all
x,y:A.x=y it is a subsingleton and we can construct a corresponding
element of Prop (aka Omega). Hence a topos corresponds to set-level HoTT
(ignoring predicativity issues for the moment).

In a language like Lean with a static universe of propositions this is not
the case. Only certain subsingletons get classified. I don't really
understand the relation but it seems that you get a quasitopos. And indeed
in this language you don't have unique choice (which was the cause of my
confusion). 

The issue was this: in a type theory with a static universe of
propositions you can add definitional proof-irrelevance, that is any two
proofs of a proposition are definitionally equal. The derivation I posted
earlier seemed to imply that you cannot have definitional
proof-irrelevance and a be in a topos. Luckily this turns out to be
incorrect and it is just due to the way proof irrelevance is implemented
in Lean (which is just wrong imho).






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