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charset=UTF-8 Content-Transfer-Encoding: quoted-printable [sorry, i always lag behind.] a global well-order assumption is not a conservative extension even of ZFC. it would make all models of computation extensional. it is also not a conservative extension of the nice non-ZF universes we like these days, including categorical, for basically the same reason. my earlier point was that, since we live among computers, the assumption of local well-ordering seems unavoidable, sooner or later. for the simple reason that all data and code are well-ordered as binaries. but the ordering is not a property, either global or local, but structure. we keep reordering things whenever we write a program. imposing a global ordering structure would require deciding what is what: identifying the intensional and the extensional equality. extensionally :) you are probably right that all this is a sort of a religious war. or maybe not a war, but rather a theological dispute. but note that theologians considered it to be self-evident that they were the bearers of the torch of logical truth. just like we. if we assume that the world is built of sets, like we were taught in math 101, then assuming well-order is indeed questionable in all kinds of ways, burdensome practically, and zermelo's solution to package it behind a big quantifier is a convenient solution. but while discussons whether the world is built of sets as the simplest foundation of everything might be interesting, or not, programmers tend to ignore them and only use sets when pedagogically unavoidable. the reason is that all types in their world are well-ordered AND that transitioning to sets and cardinalities is still as just costly as it was for cantor. so the math that runs in the computers that we use to exchange these messages, discuss foundations, and make calls home, tends to be built on well-ordered types --- which we reorder and reprogram whenever needed. intensionally :) 2c -- dusko On Mon, Feb 12, 2024 at 6:56=E2=80=AFPM Nath Rao wro= te: > [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 > mailing list group from Macquarie University. > > Leave group: > > https://url.au.m.mimecastprotect.com/s/IRyOC1WLjwsyM5vLTL23r1?domain=3Dou= tlook.office365.com > --000000000000a595c7061202ac39 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
[sorry, i always lag behind.]

a global well-order assumption is not a conservative extension even of ZF= C. it would make all models of computation extensional. it is also not a co= nservative extension of the nice non-ZF universes we like these days, inclu= ding categorical, for basically the same reason.

m= y earlier point was that, since we live among computers, the assumption of = local well-ordering seems unavoidable, sooner or later.=C2=A0 for the simpl= e reason that all data and code are well-ordered as binaries. but the order= ing is not a property, either global or local, but structure. we keep reord= ering things whenever we write a program. imposing a global ordering struct= ure would require deciding what is what: identifying the intensional and th= e extensional equality. extensionally :)

you a= re probably right that all this is a sort of a religious war. or maybe not = a war, but rather a theological dispute. but note that theologians consider= ed it to be self-evident that they were the bearers of the torch of logical= truth. just like we.

if we assume that the wo= rld is built of sets, like we were taught in math 101, then assuming well-o= rder is indeed questionable in all kinds of ways, burdensome practically, a= nd zermelo's solution to package it behind a big quantifier is a conven= ient solution.

but while discussons whether the wo= rld is built of sets as the simplest foundation of everything might be inte= resting, or not, programmers tend to ignore them and only use sets when ped= agogically unavoidable. the reason is that all types in their world are wel= l-ordered AND that transitioning to sets and cardinalities is still as just= costly as it was for cantor. so the math that runs in the computers that w= e use to exchange these messages, discuss foundations, and make calls home,= tends to be built on well-ordered types --- which we reorder and reprogram= whenever needed. intensionally :)

2c
-- dusko


On Mon, Feb 12, 2024 at 6:56=E2=80=AFPM Nath = Rao <khazanarao@gmail.com>= ; wrote:
[This n= o 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 t= he
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 t= he
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= 9;.
Then we can do global choice with no qualms.

Regards
Nath Rao





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