[Agda] How safe is Type:Type?

[Andrés Sicard-Ramírez]

On 8 March 2017 at 14:47, Andreas Abel <abela at chalmers.se> wrote:
> {-# OPTIONS --type-in-type #-} is sometimes chosen to avoid the fiddling
> with universe levels.  After all, Hurkens' Paradox is quite
> sophisticated, one should not by chance exploit the inconsistency of
> Type:Type in an innocent Agda project, no?
>
> 2. There is also Russell's paradox, written down by Paolo Capriotti.
>
>   https://github.com/agda/agda/blob/master/test/Succeed/Russell.agda

I realised Hurken's paradox doesn't use pattern-matching, that is,

  https://github.com/agda/agda/blob/aa20e7645d198de468edec62ec775ab6505b5deb/test/Succeed/Hurkens.agda

continues type-checking after adding the `--no-pattern-matching`
option. The same doesn't happen after adding that option to the above
Russell's paradox, that is, the file doesn't type-check.

1. What could be an informal description of Agda +
`--no-pattern-matching`? Agda would be something like the LF logical
framework?

2. What means that Hurkens' paradox type-checks using the
`--no-pattern-matching` option but Russell's paradox doesn't? Can I
think that Hurkens' paradox is more fundamental (whatever it means)
than Russell's paradox?

Best,
--
Andrés
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