From: Jon Sterling <j...@jonmsterling.com>
To: HomotopyTypeTheory@googlegroups.com
Subject: Re: [HoTT] Impredicative set + function extensionality + proof irrelevance consistent?
Date: Mon, 11 Dec 2017 04:43:07 -0800 [thread overview]
Message-ID: <1512996187.281015.1201017552.7F33C02E@webmail.messagingengine.com> (raw)
In-Reply-To: <aa1334b1-3b24-78d2-a670-25a27cbb186e@gmail.com>
Hi Kristina,
If you are using the subobject classifier to model Prop, I think that
you can just use the ordinary equality predicate of the topos logic. The
equality predicate for a type 'A' is given by the
(generalized/Lawvere-style) existential quantification along the
diagonal "δ : X -> X * X".
I think this ought to be the same as alternatively first constructing
the identity type in the proof-relevant fragment and then reflecting it
into the subobject classifier, since toposes model extensional identity
types.
Best,
Jon
On Mon, Dec 11, 2017, at 04:15 AM, Kristina Sojakova wrote:
> Hi Jon,
>
> Thanks for the answer! In this model, what would be the interpretation
> of the eq_A(a,b) : Prop type in Coq, which is logically equivalent to
> Id_A(a,b) : Type but proof irrelevant?
>
> Best,
>
> Kristina
>
>
> On 12/11/2017 6:42 AM, Jon Sterling wrote:
> > Can we make sense of this by interpreting into a realizability topos?
> >
> > In particular, I am under the impression that Eff (the effective topos)
> > models an impredicative universe of proof-relevant types, and on the
> > other hand its subobject classifier can be used to model a
> > proof-irrelevant Prop.
> >
> > As a topos, Eff models extensional type theory, and thence we have the
> > functional extensionality law.
> >
> > Best,
> > Jon
> >
> >
> > On Sun, Dec 10, 2017, at 08:22 PM, Kristina Sojakova 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
> >>
> >> --
> >> You received this message because you are subscribed to the Google Groups
> >> "Homotopy Type Theory" group.
> >> To unsubscribe from this group and stop receiving emails from it, send an
> >> email to HomotopyTypeThe...@googlegroups.com.
> >> For more options, visit https://groups.google.com/d/optout.
>
> --
> You received this message because you are subscribed to the Google Groups
> "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to HomotopyTypeThe...@googlegroups.com.
> For more options, visit https://groups.google.com/d/optout.
next prev parent reply other threads:[~2017-12-11 12:43 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 [this message]
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
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
Reply instructions:
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
https://en.wikipedia.org/wiki/Posting_style#Interleaved_style
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
--in-reply-to=1512996187.281015.1201017552.7F33C02E@webmail.messagingengine.com \
--to="j..."@jonmsterling.com \
--cc=HomotopyTypeTheory@googlegroups.com \
/path/to/YOUR_REPLY
https://kernel.org/pub/software/scm/git/docs/git-send-email.html
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).