From: Matt Oliveri <atm...@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: [HoTT] A puzzle about "univalent equality"
Date: Wed, 7 Sep 2016 23:34:37 -0700 (PDT) [thread overview]
Message-ID: <d53d7569-5d1f-429d-8b83-1e8c1628d8a2@googlegroups.com> (raw)
In-Reply-To: <CAOvivQxFDPaVL0iwDR=sBMWa+Zmh56jatW24-Bv7n0ar4FvaSA@mail.gmail.com>
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True, the language does not provide this structure. But that doesn't mean
it isn't in the model. Is there a problem interpreting ITT as all
Zermelo-style sets, with identity as a subsingletons for equality?
On Thursday, September 8, 2016 at 12:15:13 AM UTC-4, Michael Shulman wrote:
>
> Does "Zermelo-style set-theoretic equality" even make sense in MLTT?
> Types don't have the global membership structure that Zermelo-sets do.
>
> On Wed, Sep 7, 2016 at 4:18 PM, Matt Oliveri <atm...@gmail.com
> <javascript:>> wrote:
> > On Tuesday, September 6, 2016 at 6:14:24 PM UTC-4, Martin Hotzel Escardo
> > wrote:
> >>
> >> Some people like the K-axiom for U because ... (let them fill the
> answer).
> >
> >
> > It allows you to interpret (within type theory) the types of ITT + K as
> > types of U, and their elements as the corresponding elements.
> (Conjecture?)
> > Whereas without K, we don't know how to interpret ITT types as types of
> U.
> > In other words, K is a simple way to give ITT reflection.
> >
> >> Can we stare at the type (Id U X Y) objectively, mathematically, say
> >> within intensional MLTT, where it was introduced, and, internally in
> >> MLTT, ponder what it can be, and identify the only "compelling" thing
> it
> >> can be as the type of equivalences X~Y, where "compelling" is a notion
> >> remote from univalence?
> >
> >
> > How does Zermelo-style set-theoretic equality get ruled out as a
> potential
> > "compelling" meaning for identity? Of course, there's a potential
> argument
> > about what "compelling" ought to be getting at. You seem to not consider
> > set-theoretic equality compelling, which I can play along with.
>
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next prev parent reply other threads:[~2016-09-08 6:34 UTC|newest]
Thread overview: 18+ messages / expand[flat|nested] mbox.gz Atom feed top
2016-09-05 16:54 Andrew Polonsky
2016-09-05 21:40 ` [HoTT] " Michael Shulman
2016-09-05 21:51 ` Dan Licata
2016-09-06 7:30 ` Andrew Polonsky
2016-09-06 12:32 ` Michael Shulman
2016-09-06 12:56 ` Dan Licata
2016-09-06 12:57 ` Peter LeFanu Lumsdaine
2016-09-06 13:44 ` Andrew Polonsky
2016-09-06 22:14 ` Martin Escardo
2016-09-07 23:18 ` Matt Oliveri
2016-09-08 4:14 ` Michael Shulman
2016-09-08 6:06 ` Jason Gross
2016-09-08 9:11 ` Martin Escardo
2016-09-08 6:34 ` Matt Oliveri [this message]
2016-09-08 6:45 ` Michael Shulman
2016-09-08 9:07 ` Martin Escardo
2016-09-08 9:51 ` Thomas Streicher
2016-09-19 12:40 ` Robin Adams
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