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T= he contradiction is described in Theorem 3.2.2 of the HoTT Book. Of course, one is free to use traditional set theoretic foundations (either= in terms of ZFC, or in terms of its embedding into UF + choice via iterati= ve notions of set), and in this case the usual classical results about the = consistency of indefinite descriptions would be available =E2=80=94 but the= reason I bring this up is that univalent foundations offers a compelling a= lternative to the usual approach of filling in missing data via choice that= you bring up. In set theoretic foundations, the missing data is not unique= ly determined (it is only unique up to isomorphism) and so you need some ki= nd of choice to get it back; but in univalent foundations, it very often (i= n fact, usually!) happens that this missing data is uniquely determined in = the appropriate sense =E2=80=94 in this way, univalence increases the scope= of *definite* descriptions so much that many classical uses of choice and/= or indefinite descriptions become obsolete. This is why, in univalent found= ations, you can avoid choosing a cleaving of a fibration because all fibrat= ions are automatically, canonically, cloven. It makes sense that there is a variety of approaches to deal with such issu= es. Best, Jon On Mon, Feb 12, 2024, at 1:20 PM, Nath Rao wrote: > [This no longer about how to define fibrewise opposite fibration, so may > be it is time t change the subject line.] > > IMHO, this seems to be more like a religious war. In "mathematics in the > small", local axiom of choice allows us to choose representatives only > when needed. So we are grew up believing that putting things in the > definition that are not preserved by morphisms is "not done" (the > religious dogma) . But them we run up against issues when we try to do > the same thing in "mathematics in the large", and either ignore the > problem or argue endlessly about how to solve it. > > Also, if are willing to assume that every set comes with a > well-ordering, why not assume a global well-ordering and be done with > it? [If I am going to use universes anyway, it seems to be trivial to do > this, as long as I live inside a fixed universe in any given proof.] For > ZFC, global choice is a conservative extension. Not being a logician, am > not sure, but perhaps it can be shown for global choice is a > conservative extension of local choice of 'all suitable set theories'. > Then we can do global choice with no qualms. > > Regards > Nath Rao > > > > > > ---------- > > You're receiving this message because you're a member of the Categories= =20 > mailing list group from Macquarie University. > > Leave group: > https://protect-au.mimecast.com/s/Uy0cCp81gYC8WpnQSP-lCQ?domain=3Doutlook= .office365.com --MCBoundary=_12402131916440121 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=utf-8 Dear Nath,

It is important to note that although global choice is a very useful tool i= ndeed, it is not consistent with the interpretation of the universe of sets= in univalent foundations (UF) =E2=80=94 despite the fact that the usual ax= iom of choice can be added to univalent foundations without any problems. T= he contradiction is described in Theorem 3.2.2 of the HoTT Book.

Of course, one is free to use traditional set theoretic foundations (either= in terms of ZFC, or in terms of its embedding into UF + choice via iterati= ve notions of set), and in this case the usual classical results about the = consistency of indefinite descriptions would be available =E2=80=94 but the= reason I bring this up is that univalent foundations offers a compelling a= lternative to the usual approach of filling in missing data via choice that= you bring up. In set theoretic foundations, the missing data is not unique= ly determined (it is only unique up to isomorphism) and so you need some ki= nd of choice to get it back; but in univalent foundations, it very often (i= n fact, usually!) happens that this missing data is uniquely determined in = the appropriate sense =E2=80=94 in this way, univalence increases the scope= of *definite* descriptions so much that many classical uses of choice and/= or indefinite descriptions become obsolete. This is why, in univalent found= ations, you can avoid choosing a cleaving of a fibration because all fibrat= ions are automatically, canonically, cloven.

It makes sense that there is a variety of approaches to deal with such issu= es.

Best,
Jon


On Mon, Feb 12, 2024, at 1:20 PM, Nath Rao wrote:
> [This no longer about how to define fibrewise opposite fibration, so m= ay
> be it is time t change the subject line.]
>
> IMHO, this seems to be more like a religious war. In "mathematics= in the
> small", local axiom of choice allows us to choose representatives= only
> when needed. So we are grew up believing that putting things in the > definition that are not preserved by morphisms is "not done"= (the
> religious dogma) . But them we run up against issues when we try to do=
> the same thing in "mathematics in the large", and either ign= ore the
> problem or argue endlessly about how to solve it.
>
> Also, if are willing to assume that every set comes with a
> well-ordering, why not assume a global well-ordering and be done with<= BR> > it? [If I am going to use universes anyway, it seems to be trivial to = do
> this, as long as I live inside a fixed universe in any given proof.] F= or
> ZFC, global choice is a conservative extension. Not being a logician, = am
> not sure, but perhaps it can be shown for global choice is a
> conservative extension of local choice of 'all suitable set theories'.=
> Then we can do global choice with no qualms.
>
> Regards
> Nath Rao
>
>
>
>
>
> ----------
>
> You're receiving this message because you're a member of the Categorie= s
> mailing list group from Macquarie University.
>
> Leave group:
> https://outlook.office365.com/owa/categories@= mq.edu.au/groupsubscription.ashx?source=3DEscalatedMessage&action=3Dlea= ve&GuestId=3D3d33085c-21ea-4036-99e4-8fc96e8c25b8
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