From: Andrew Polonsky <andrew....@gmail.com>
To: Thomas Streicher <stre...@mathematik.tu-darmstadt.de>
Cc: Andrej Bauer <andrej...@andrej.com>,
Martin Escardo <escardo...@googlemail.com>,
"HomotopyT...@googlegroups.com" <HomotopyT...@googlegroups.com>
Subject: Re: [HoTT] Meta-conjecture about MLTT
Date: Mon, 12 Sep 2016 13:23:09 +0200 [thread overview]
Message-ID: <CABcT7WDW4nj7kpeoEoTS0nB5X0MB04ZCf8t3g027=NE9H9NVOg@mail.gmail.com> (raw)
In-Reply-To: <20160912111420.GA14499@mathematik.tu-darmstadt.de>
Perhaps one does not need to take a skeleton.
A dependent type x:U |- P(X) : Type gives rise, in the *standard*
set-theoretic model, to a family of sets indexed by [[U]].
Since the standard type constructor preserve isomorphism, two fibers
[[P(A)]] and [[P(B)]] will be isomorphic if A and B are.
Saying "A provably has property P" translates into "[[P(A)]] is
inhabited by a definable element".
In principle, one could conceive that [[P(B)]] has no definable
elements, even if the two are isomorphic.
But saying that "B provably does not have property P" translates into
"[[P(B)]] is empty", which cannot be the case.
Cheers,
Andrew
On Mon, Sep 12, 2016 at 1:14 PM, Thomas Streicher
<stre...@mathematik.tu-darmstadt.de> wrote:
> I am afraid this skeleton model is not split and spliiting it it wan't
> be a skeleton anymore.
>
> Thomas
>
>
> On Mon, Sep 12, 2016 at 01:02:24PM +0200, Andrej Bauer wrote:
>> Let me try.
>>
>> Consider the model of MLTT in the skeleton of sets, i.e., types are
>> interpreted as cardinal numbers and functions are the set-theoretic
>> functions between cardinals. The identity type is just what you'd
>> expect in a set-theoretic model (in fact, there is *no* choice about
>> i). There are no problems of coherence anywhere.
>>
>> This model validates equality reflection. If MLTT+reflection
>> distinguished isomorphic types, then there would exist in this model
>> two isomorphic types which are not equal, but since we started with a
>> skeleton this will not do.
>>
>> It is a bit amusing that the model also validates things like Id(U,
>> Nat -> Nat, Nat -> Bool) and is thus quite educational.
>>
>> It is a model, right?
>>
>> With kind regards,
>>
>> Andrej
>>
>
> --
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next prev parent reply other threads:[~2016-09-12 11:23 UTC|newest]
Thread overview: 17+ messages / expand[flat|nested] mbox.gz Atom feed top
2016-09-11 20:47 Martin Escardo
2016-09-11 22:08 ` [HoTT] " Peter LeFanu Lumsdaine
2016-09-11 22:26 ` Martin Escardo
2016-09-12 9:11 ` Thomas Streicher
2016-09-12 5:20 ` Andrew Polonsky
2016-09-12 5:59 ` [HoTT] " Jason Gross
2016-09-12 9:55 ` Andrew Polonsky
2016-09-12 10:07 ` Andrew Polonsky
2016-09-12 10:35 ` Nicolai Kraus
2016-09-12 10:16 ` Peter LeFanu Lumsdaine
2016-09-12 10:44 ` [HoTT] " Nicolai Kraus
2016-09-12 11:02 ` Andrej Bauer
2016-09-12 11:14 ` Thomas Streicher
2016-09-12 11:23 ` Andrew Polonsky [this message]
2016-09-12 11:41 ` Thomas Streicher
2016-09-12 12:47 ` Thomas Streicher
2016-09-12 13:01 ` Martin Escardo
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