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 >> wrote: >> >> See https://www.nature.com/articles/d41586-017-05477-9 >> >> -- >> 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. >> >> >>