From: Andrea Vezzosi <sanz...@gmail.com>
To: Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>
Cc: Kristina Sojakova <sojakova...@gmail.com>,
Homotopy Type Theory <homotopyt...@googlegroups.com>
Subject: Re: [HoTT] Impredicative set + function extensionality + proof irrelevance consistent?
Date: Tue, 12 Dec 2017 11:15:48 +0100 [thread overview]
Message-ID: <CAOSJkmwoF3pFLw+YpqS5QKE3eNbOf2v2TyURsVx9AGK-RwygZg@mail.gmail.com> (raw)
In-Reply-To: <BB338612-ECF6-42F8-B672-6276F5D7BCE8@exmail.nottingham.ac.uk>
On Mon, Dec 11, 2017 at 3:23 PM, Thorsten Altenkirch
<Thorsten....@nottingham.ac.uk> wrote:
> Hi Kristina,
>
> I guess you are not assuming Prop:Set because that would be System U and hence inconsistent.
>
> By proof-irrelevance I assume that you mean that any two inhabitants of a proposition are definitionally equal. This assumption is inconsistent with it being a tops since in any Topos you get propositional extensionality, that is P,Q : Prop, (P <-> Q) <-> (P = Q), which is indeed an instance of univalence.
>
I don't know if it's relevant to the current discussion, but I thought
the topos of sets with Prop taken to be the booleans would support
both proof irrelevance and propositional extensionality, classically
at least. Is there some extra assumption I am missing here?
> It should be possible to use a realizability semantics like omega-sets or Lambda-sets to model the impredicative theory and identify the propositions with PERs that are just subsets.
>
> Cheers,
> Thorsten
>
>
> On 11/12/2017, 04:22, "homotopyt...@googlegroups.com on behalf of Kristina Sojakova" <homotopyt...@googlegroups.com on behalf of sojakova...@gmail.com> wrote:
>
> Dear all,
>
> I asked this question last year on the coq-club mailing list but did not
> receive a conclusive answer so I am trying here now. Is the theory with
> a proof-relevant impredicative universe Set, proof-irrelevant
> impredicative universe Prop, and function extensionality (known to be)
> consistent? It is known that the proof-irrelevance of Prop makes the Id
> type behave differently usual and in particular, makes the theory
> incompatible with univalence, so it is not just a matter of tacking on
> an interpretation for Prop.
>
> Thanks in advance for any insight,
>
> Kristina
>
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next prev parent reply other threads:[~2017-12-12 10:15 UTC|newest]
Thread overview: 54+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-12-11 4:22 Kristina Sojakova
2017-12-11 11:42 ` [HoTT] " Jon Sterling
2017-12-11 12:15 ` Kristina Sojakova
2017-12-11 12:43 ` Jon Sterling
2017-12-11 14:28 ` Thomas Streicher
2017-12-11 14:32 ` Kristina Sojakova
2017-12-11 14:23 ` Thorsten Altenkirch
2017-12-12 10:15 ` Andrea Vezzosi [this message]
2017-12-12 11:03 ` Thorsten Altenkirch
2017-12-12 12:02 ` Thomas Streicher
2017-12-12 12:21 ` Thorsten Altenkirch
2017-12-12 13:17 ` Jon Sterling
2017-12-12 19:29 ` Thomas Streicher
2017-12-12 19:52 ` Martin Escardo
2017-12-12 23:14 ` Michael Shulman
2017-12-14 12:32 ` Thorsten Altenkirch
2017-12-14 18:52 ` Michael Shulman
2017-12-16 15:21 ` Thorsten Altenkirch
2017-12-17 12:55 ` Michael Shulman
2017-12-17 17:08 ` Ben Sherman
2017-12-17 17:16 ` Thorsten Altenkirch
2017-12-17 22:43 ` Floris van Doorn
2017-12-15 17:00 ` Thomas Streicher
2017-12-17 8:47 ` Thorsten Altenkirch
2017-12-17 10:21 ` Thomas Streicher
2017-12-17 11:39 ` Thorsten Altenkirch
2017-12-18 7:41 ` Matt Oliveri
2017-12-18 10:00 ` Michael Shulman
2017-12-18 11:55 ` Matt Oliveri
2017-12-18 16:24 ` Michael Shulman
2017-12-18 20:08 ` Matt Oliveri
2017-12-18 10:10 ` Thorsten Altenkirch
2017-12-18 11:17 ` Matt Oliveri
2017-12-18 12:09 ` Matt Oliveri
2017-12-18 11:52 ` Thomas Streicher
2017-12-19 11:26 ` Thorsten Altenkirch
2017-12-19 13:52 ` Andrej Bauer
2017-12-19 14:44 ` Thorsten Altenkirch
2017-12-19 15:31 ` Thomas Streicher
2017-12-19 16:10 ` Thorsten Altenkirch
2017-12-19 16:31 ` Thomas Streicher
2017-12-19 16:37 ` Thorsten Altenkirch
2017-12-20 11:00 ` Thomas Streicher
2017-12-20 11:16 ` Thorsten Altenkirch
2017-12-20 11:41 ` Thomas Streicher
2017-12-21 0:42 ` Matt Oliveri
2017-12-22 11:18 ` Thorsten Altenkirch
2017-12-22 21:20 ` Martín Hötzel Escardó
2017-12-22 21:36 ` Martín Hötzel Escardó
2017-12-23 0:25 ` Matt Oliveri
2017-12-19 16:41 ` Steve Awodey
2017-12-20 0:14 ` Andrej Bauer
2017-12-20 3:55 ` Steve Awodey
[not found] ` <fa8c0c3c-4870-4c06-fd4d-70be992d3ac0@skyskimmer.net>
2017-12-14 13:28 ` Thorsten Altenkirch
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