Re: Getting rid of cardinality as an issue

Re: Getting rid of cardinality as an issue

[Vaughan Pratt]

>From: Mike Oliver <moliver@unt.edu>
>Surely, from time to time,
>categorists must care about genuinely ultra-first-order notions, such as
>(say) the metric completeness of the real numbers?  To me the natural
>way of getting such notions right is to make sure that each of your
>universes is closed under the (true) powerset operation.

Yes, which was why I formulated the exponential gap only as a lower bound.
The idea was that if you needed more, take it.

In retrospect I should have included the Russell paradox, viewed
constructively as a set factory rather than mysteriously as a bogeyman
under the bed, as something one might or might not need for some purpose,
e.g. as a successor function.  This reclassification (as a shift only in my
personal outlook) prompts me to withdraw my suggestion (made as much for
my benefit as anyone's but as such good to bounce off people) of imposing
any lower bound at all on size of gaps between successive n-CAT categories.

Size can certainly be an issue, whether involving rates of growth of functions
on the integers, or large cardinals.  In their CM104 book, Makkai and Pare
treat the first order model theory (as opposed to first order logic) of
accessible categories, where the goal is to characterize the behavior of
categories independently of their size as far as possible, and where not
possible to characterize the dependencies on size.  Such an enterprise is
not ordinary mathematics but foundations, and as such is *about* these gaps.
Their results (presumably with the help of a consultant) should allow those
on the consuming side of foundations, i.e. those doing ordinary mathematics
(if there really is such a thing), to judge for themselves whether a given
construction is in danger of colliding with a size paradox.  One would hope
that a few simple rules of thumb would minimize dependence on consultants,
though it did not seem to me that CM104 was organized with that economy
clearly in mind; this might be corrected with a short cheat sheet as an
addendum.

Not all paradoxes concern size.  The liar paradox and the
division-by-infinitesimal paradoxes can be turned into size paradoxes via
a suitable encoding, but they are not intrinsically size-related; well,
in the case of infinitesimals, not large sizes anyway.

Perhaps it just reflects my old-fashioned upbringing, but the foundational
role intended for CM104 is way clearer to me than any of the several topos
texts currently scattered around my desk.  Not with regard to the definitions,
examples, and (to the extent I understand their motivation) the theorems
of topos theory.  The elementary definition of a topos is crystal clear
(not to mention incredibly beautiful), as are the basic examples of toposes.

Where I run into problems is in placing topos theory as a foundation beside
say accessible categories.  I can go repeatedly through the topos texts
and just not get it.  Is there some finely honed sentence or paragraph that
explains this relationship?

I get the feeling there should be a sentence or paragraph to the effect that
one brings size under control (or makes it a non-issue) by passing from the
external logic of accessible categories to the internal logic of toposes.
Is some such clear and succinct story (not necessarily that one since it
might be totally wrong) told somewhere?  If so, one could deal with idiots
like me who rant about size as an issue by pointing them at that story,
by way of indicating how to stop worrying about inaccessible cardinals
by embracing someone else's internal logic (and making it one's own?).
Or whatever the story actually is.

What about Remark 7.1.14 in Paul Taylor's Practical Mathematics, for example?
Is this tangential, on point, or core?  What about the preface to Borceux'
Volume 3?  Does Peter Johnstone's nonconstructive theorem "There exists
an elephant" in his preface have a succinctly summarized constructive
counterpart somewhere, a sort of sharply focused photo of an elephant taken
from 50 feet away?  (Actually I suppose a sharply focused photo of a real
elephant would have very close to the same number of megabytes of data as
in the two volumes, so maybe I mean an elephant icon.)

Or is this all just a misunderstanding or misinterpretation of the real
goals of topos theory, with the truth being that there is ultimately no
way mathematicians can avoid large cardinals if they expect to be able to
prove certain theorems, even those of an ostensibly combinatorial flavor?
This is certainly the sermon that Harvey Friedman has been preaching for
a number of years; is Harvey wrong about this?

There seem to be some sunglasses and rose-colored glasses lying around but
I can't tell who they belong to.  Surely they're not all mine.

Vaughan Pratt

Re: Getting rid of cardinality as an issue

[Mike Oliver]

Vaughan Pratt wrote:

> While I'm happy to field objections like "too flippant", I'm more concerned as
> to whether there are any technical flaws, and to a lesser extent philosophical
> or religious concerns.  (I would not want to be held responsible for guns
> being brought to the next UACT meeting if ever there is one.)
> [...]
> Now the Grothendieck hierarchy is stepped through via ZF rather than Z, with
> Fraenkel's Replacement axiom doing the heavy hitting.  This creates
gaps
> mind-bogglingly larger than my teensy exponential gaps above.  The general
> idea seems to be that these gaps ought to be large enough to take care of
> Russell while still not running headlong into inconsistency.  However gaps
> this large do entail a certain amount of finger-crossing, and one might
> question the logic of hitting Russell with a nuclear weapon that might send
> some fallout your way when a harmless little tack-hammer will take him out.

I'm not entirely sure I follow what Vaughan's project is here, so this
may come out as a non sequitur, but:  Surely, from time to time,
categorists must care about genuinely ultra-first-order notions, such as
(say) the metric completeness of the real numbers?  To me the natural
way of getting such notions right is to make sure that each of your
universes is closed under the (true) powerset operation.  That would
require the cardinality of your universes to be, at least, strong limit
cardinals.

Having them closed under ranges of functions also seems natural enough;
at that point you need inaccessibles.

It's by no means clear that inaccessibles are sufficient.  What happens
when you want to be closed under the operation of finding the next
larger inaccessible?

Re: Getting rid of cardinality as an issue

[Dusko Pavlovic]

i think the question of foundations needs to be considered together with
the meta-question: why working mathematicians don't care for foundations?

a trivial part of the answer is that it's a matter of taste: some people
organize their diet following the pyramid of "so much fruit so much
vegetables so much meat", other people smoke and drink coffee and eat
chocolate.

the less trivial part of the answer is that the world of working
mathematics is not built on top of a static foundation. the questions
and the meta-questions are asked together. categories are foundations of
categories.

russell's paradox and hilbert's idea that math should have a static
foundation are old. a lot has happened. sets are not so rigid any more.
starting from models of untyped lambda calculus, people built all kinds
of reflective universes, even containing small complete categories. the
category of small categories can probably be a small 2-category in such
a universe.

the set of all sets can hardly be a set because of the variance, but i
think that the set of all sets of sets can be a set in some models.

my 2p,
-- dusko


Vaughan Pratt wrote:

>Encouraged by the lack of objections to my previous message about why
>Russell's Paradox should not be a big deal, I had a shot at shrinking the
>position I spelled out there down to one paragraph, as follows.
>
>------------
>We shall axiomatize certain 1-categories using 2-categories.  We avoid
>Russell's paradox by treating any aggregation of $n$-categories as an
>$(n+1)$-category, and allowing for the possibility that the
>$(n+1)$-category
>$n$-$\CAT$ of all $n$-categories might be exponentially larger than any of
>its members.  We impose no other size constraints besides the obvious
>one of keeping things small enough to remain consistent.  Sets are defined
>as usual as 0-categories and categories as 1-categories.
>------------
>
>While I'm happy to field objections like "too flippant", I'm more concerned as
>to whether there are any technical flaws, and to a lesser extent philosophical
>or religious concerns.  (I would not want to be held responsible for guns
>being brought to the next UACT meeting if ever there is one.)
>
>Makkai and Pare address the same issue in AMS CM 104 (Accessible Categories)
>with a hierarchy of Grothendieck universes (three, since they like me stop at
>2-categories for the application at hand).
>
>Now the Grothendieck hierarchy is stepped through via ZF rather than Z, with
>Fraenkel's Replacement axiom doing the heavy hitting.  This creates gaps
>mind-bogglingly larger than my teensy exponential gaps above.  The general
>idea seems to be that these gaps ought to be large enough to take care of
>Russell while still not running headlong into inconsistency.  However gaps
>this large do entail a certain amount of finger-crossing, and one might
>question the logic of hitting Russell with a nuclear weapon that might send
>some fallout your way when a harmless little tack-hammer will take him out.
>
>One objection I can readily imagine to the above is that I've conflated
>the n-category hierarchy with Russell's proposal for a ramified types
>hierarchy.  I would disagree with that.  All I have done is to insist
>on two things that seem to me to be independent.
>
>1.  I have proposed to call aggregations of n-categories (n+1)-categories.
>Now morphisms between n-categories are n-functors, and where there are
>n-functors there are n-natural transformations, so this is hardly a bold
>proposal.
>
>2.  *Some* gap is needed between n-CAT and (n+1)-CAT, starting with the
>requirement that Set be bigger than any set.  Russell's paradox is no
>respecter of n, applying just as effectively to an (n+1)-category of
>n-categories as it does to a 1-category of sets.
>
>Certainly I have juxtaposed 1 and 2, but that is not the same thing as
>conflating them.  Their mere juxtaposition provides sufficient armor
>against both Russell's paradox and the Icarus risk of flying too close to
>an inconsistently large cardinal.
>
>The "prior art" for dealing with these issues has given rise to the adjectives
>"small", "large," "superlarge", etc. and the nouns "set" and "class."
>A good test for any revolution is the amount of blood it needs to shed.
>The following definitions are aimed at minimal upheaval through maximum
>compatibility with the status quo.
>
>* An object is n-small when it belongs to an n-category.
>
>* Small = 1-small, large = 2-small, superlarge = 3-small, etc.
>
>* A set is a discrete 1-category.
>
>* A class is a discrete n-category for unspecified n.
>
>Hopefully Sol Feferman will give an even simpler solution in his talk
>tomorrow.
>
>Vaughan Pratt
>
>
>
>
>
>

Getting rid of cardinality as an issue

[Vaughan Pratt]

Encouraged by the lack of objections to my previous message about why
Russell's Paradox should not be a big deal, I had a shot at shrinking the
position I spelled out there down to one paragraph, as follows.

------------
We shall axiomatize certain 1-categories using 2-categories.  We avoid
Russell's paradox by treating any aggregation of $n$-categories as an
$(n+1)$-category, and allowing for the possibility that the
$(n+1)$-category
$n$-$\CAT$ of all $n$-categories might be exponentially larger than any of
its members.  We impose no other size constraints besides the obvious
one of keeping things small enough to remain consistent.  Sets are defined
as usual as 0-categories and categories as 1-categories.
------------

While I'm happy to field objections like "too flippant", I'm more concerned as
to whether there are any technical flaws, and to a lesser extent philosophical
or religious concerns.  (I would not want to be held responsible for guns
being brought to the next UACT meeting if ever there is one.)

Makkai and Pare address the same issue in AMS CM 104 (Accessible Categories)
with a hierarchy of Grothendieck universes (three, since they like me stop at
2-categories for the application at hand).

Now the Grothendieck hierarchy is stepped through via ZF rather than Z, with
Fraenkel's Replacement axiom doing the heavy hitting.  This creates gaps
mind-bogglingly larger than my teensy exponential gaps above.  The general
idea seems to be that these gaps ought to be large enough to take care of
Russell while still not running headlong into inconsistency.  However gaps
this large do entail a certain amount of finger-crossing, and one might
question the logic of hitting Russell with a nuclear weapon that might send
some fallout your way when a harmless little tack-hammer will take him out.

One objection I can readily imagine to the above is that I've conflated
the n-category hierarchy with Russell's proposal for a ramified types
hierarchy.  I would disagree with that.  All I have done is to insist
on two things that seem to me to be independent.

1.  I have proposed to call aggregations of n-categories (n+1)-categories.
Now morphisms between n-categories are n-functors, and where there are
n-functors there are n-natural transformations, so this is hardly a bold
proposal.

2.  *Some* gap is needed between n-CAT and (n+1)-CAT, starting with the
requirement that Set be bigger than any set.  Russell's paradox is no
respecter of n, applying just as effectively to an (n+1)-category of
n-categories as it does to a 1-category of sets.

Certainly I have juxtaposed 1 and 2, but that is not the same thing as
conflating them.  Their mere juxtaposition provides sufficient armor
against both Russell's paradox and the Icarus risk of flying too close to
an inconsistently large cardinal.

The "prior art" for dealing with these issues has given rise to the adjectives
"small", "large," "superlarge", etc. and the nouns "set" and "class."
A good test for any revolution is the amount of blood it needs to shed.
The following definitions are aimed at minimal upheaval through maximum
compatibility with the status quo.

* An object is n-small when it belongs to an n-category.

* Small = 1-small, large = 2-small, superlarge = 3-small, etc.

* A set is a discrete 1-category.

* A class is a discrete n-category for unspecified n.

Hopefully Sol Feferman will give an even simpler solution in his talk
tomorrow.

Vaughan Pratt