[Agda] Forget Hurken's Paradox, Agda has a quicker route to success

Altenkirch Thorsten psztxa at exmail.nottingham.ac.uk
Mon Sep 17 11:06:00 CEST 2012

On 16/09/2012 21:09, "Dan Doel" <dan.doel at gmail.com> wrote:

>On Thu, Sep 6, 2012 at 12:47 PM, Nils Anders Danielsson <nad at chalmers.se>
>> On 2012-09-06 17:21, Andreas Abel wrote:
>>> The problem is that Agda considers
>>>    F D = \Sigma E : Set. (D ≡ E) * (E → ⊥)
>>> as strictly positive in D.
>> I'm trying to understand if this is actually a problem. Are there any
>> dire consequences if we stick to predicative types?

I am concerned what happens on the next level, I.e. can we construct a
fixpoint of

F : Set₁ → Set₁
F X = Σ[ E ː Set ] (X ≡ lift E × (E → ⊥))


Or alternatively we could use a universe (U : Set, El U -> Set):

F : Set → Set
F X = Σ[ E ː U ] (X ≡ El E × (E → ⊥))

That looks dangerous…


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