[Agda] Classical Mathematics for a Constructive World

[Altenkirch Thorsten]

>Notnot appears to be a (crude?) form of "squashing", or "bracketing", or
>"smashing" constructions
>of a Sigma type down to "just one", or hiding all distinctions between
>them.
>It seems to have a quite dramatic effect on principles like CT,
>continuity, and even choice.
>Maybe there are other forms of squashing.

Indeed, there is the canonical _* : Set -> Prop which reflects Set into
Prop (can be also read as "inhabited"). It is equivalent to the quotient
with the "chaotic" equivalence relation (always true) under the priviso
that the by Set we mean hSet.

Hence we can express

Another alternative would be to use

WhCT: (Pi f : N->N)[(Sigma e : N)(Pi n : N)(Sigma m : N) T(e,n,m) and
f(n)=U(m)]*

which we could also write as

WhCT: (Pi f : N->N)[(Exists e : N)(Pi n : N)(Sigma m : N) T(e,n,m) and
f(n)=U(m)]

where Exists A P = (Sigma A P)*. How would this affect the results you
mention?


Thorsten

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