Re: [HoTT] Voevodsky obituary in Nature

["Martín Hötzel Escardó" <"escardo..."@gmail.com>]

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On Tuesday, 7 November 2017 22:49:30 UTC, Martín Hötzel Escardó wrote:
>
>
>
> On Tuesday, 7 November 2017 21:50:24 UTC, Dimitris Tsementzis wrote:
>>
>> Dan,
>>
>> It is an interesting idea to refer to univalence as a “mechanism” rather 
>> than an “axiom”.
>>
>> Even if you did it for expository reasons, I believe that in certain 
>> contexts it is a good phrase to adopt.
>>
>
> I very much like this way of expression, too, even though univalence 
> became a mechanism for the first time with cubicaltt.
>
> Nevertheless, although Vladimir certainly didn't have a mechanism for 
> univalence, he envisioned its possibility, and even formulated a conjecture 
> that is still unsettled, namely that if univalence implies that there is a 
> *number* n:N with a property P(n), then we can find, metatheoretically, 
> using an algorithm, a *numeral* n':N (a closed term of type N) and a proof 
> that univalence implies P(n'). 
>

The conjecture said that we can find a numeral n':N and a proof that 
univalence implies n=n' . 

Here is what he actually conjectured:

"
Conjecture 1 There is a terminating algorithm which for any term nn of type 
nat constructed
with the use of the univalence axiom outputs a pair (nn 0 , pf ) where nn 0 
is a term of type nat
constructed without the use of the univalence axiom and pf is a term of 
type paths nat nn nn 0 (i.e.
a proof that nn 0 = nn). The term pf may use the univalence axiom.
"
 http://www.math.ias.edu/vladimir/files/univalent_foundations_project.pdf

Martin

No mechanism for univalence can currently provide that, as far as I know.
>
> Martin 
>  
>
>>
>> Dimitris
>>
>> On Nov 7, 2017, at 16:40, Daniel R. Grayson <daniel...@gmail.com> 
>> wrote:
>>
>> See https://www.nature.com/articles/d41586-017-05477-9
>>
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