Getting rid of cardinality as an issue

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

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
>
>
>
>
>
>

Re: Re: Getting rid of cardinality as an issue

[Eduardo Dubuc]

Dusko Pavlovic wrote:

"why working mathematicians don't care for foundations?"

very simple anwer:

foundations is just an area within mathematics

the working mathemeticians who care about foundations are those who work
in foundations

why do not ask the question:

"why working mathematicians don't care for ring theory?"

well, because we think that those who work in ring theory are working
mathematicians

but there are a lot who do not work in ring theory, and do not care
either

for foundations it is the same thing

why we give foundations a different status ?

saludos  eduardo dubuc

Re: Getting rid of cardinality as an issue (correction)

[Toby Bartels]

Vaughan Pratt wrote:

>[Note from moderator: apologies to Vaughan for missing his requested
>change: 1 has been changed to 0 5 lines from bottom, so it reads:
>`discrete 0-category'.]

And this is the line in question:

>* A set is a discrete 0-category.

Just to check, the word "discrete" here is redundant, right?
You just put it in to contrast with the next line, where it's necessary:

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


-- Toby

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 (correction)

[Vaughan Pratt]

>[Note from moderator: apologies to Vaughan for missing his requested
>change: 1 has been changed to 0 5 lines from bottom, so it reads:
>`discrete 0-category'.]

Just for the record, the change I requested was from "discrete
1-category" to just "0-category" without the "discrete" (being redundant).

Vaughan