1 Jul 08:55 2011

### CT2011-registration


The CT2011 website

http://www.mat.uc.pt/~ct2011/index.html

now contains a list consisting of registered participants as well as
a list of those who will participate but have not yet registered or whose
registration is currently in process.

If you are not on either of those lists and intend to participate, then
please let me know of your intention by July 4 so that there will be space
saved for you at the banquet and excursion.

that information to me.

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2 Jul 14:24 2011

### Re: size_question

On Wed, Jun 29, 2011 at 4:43 AM, James Lipton <jlipton <at> wesleyan.edu> wrote:
> These are the "hereditarily finite sets"
>
>    V(0) = empty set
>
>   V(n+1) = P(V(n))
>
> V(omega) = U{V(n): n in omega}
>
> I would not call it "the" cat of finite sets, since there uncountably many
> countable models of ZF. Obviously the choice of ZF, rather than say Zermelo
> set theory or some other foundation is also pretty arbitrary.

I should think that the hereditarily finite sets do not depend all
that much on the background setting. After all, there are not very
many of them and they are quite concrete. Can they really be hugely
different depending on whether we work in ZF, ZFC, IZF, CZF etc?

With kind regards,

Andrej

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2 Jul 22:05 2011

### Re: size_question

> I should think that the hereditarily finite sets do not depend all
> that much on the background setting. After all, there are not very
> many of them and they are quite concrete. Can they really be hugely
> different depending on whether we work in ZF, ZFC, IZF, CZF etc?

If we are not working classically subsets of finite sets need not be
finite but the sets in V_\omega are closed under subsets.

Thomas

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3 Jul 17:01 2011

### Re: size_question


In the absence of AC, we need to specify which notions of"finite" we are using. Certainly K-S (= locally
quotient of standard numeral) while important is by no means the endof the story. We categorists do not
seem to yet have a way of dealing  elegantly with the locally Noetherian, coherent,etcsheaves in terms of
internal finiteness notions.
The SUBQUOTIENTS of standard numerals would surely be important. I recall the existence of intuitionist
literature on this, but it seemed to assume that subquots of subquots etc would be an infinite sequence. Of
course in a  category with reasonable pullbacks and pushouts this relation is already transitive (indeed
an important subcategory of spans).
But the original question actually had to do with the observation that for any qualitative definition of
finite set, there are probably as many as there things in the ambient universe. We need an axiom of infinity
to say that there is a category object that represents that metacategory UP TO EQUIVALENCE OF COURSE.
The uniqueness is only relative to the ambient universe . The Incompleteness Theorem would seem to imply
that there are an infinite number of non-elementarily-equivalent ambient universesand hence of these
little metacategories in particular.
We really should overcome the ritual belief that such things must be defined by iteration (as opposed to
being partially investigated via iteration). Already Peano misrepresented Grassmann's views on this.
Dedekind proposed that a set  A is finite if any idempotent whose fixed part is isomorphic to all of A is itself
an automorphism. This seems difficultto relate to operations such as product. However note that it is an
elementary axiom to require that all objects of a given topos satisfy it. It would see to propagate to any
finite (in the sense of Artin) topos over such. Do basictheorems, such as the essentialness  of all
geometric morphisms, extend to this axiomatic setting ?
A different elementary axiom on a topos is the requirement that every object A is fixed by the monad obtained
by composing 3^( ) with its adjoint from the related topos of left actions of 3^3.
Are these two theories equivalent ? Such a finitely-axiomatized T defines "the" category of finite sets as
any one that represents up to equivalence the submetacategory of the ambient universe consisting of all
discrete categories that are finite in the stated sense.( But then of course there are many  models of T not
equivalent to that)
I proposed the study of such an "Objective Number Theory" in Braga four years ago at a European computer
science meeting without much response. It is known since Craig-Vaught that Peano's arithmetic can be


4 Jul 00:59 2011

Thanks to all who wrote something on this question.

It clarified mi ignorance:

There is the category of finite sets, namely, the category of all those
sets which happen to be finite. No need of more precision. But it is not
small (or an element of the universe if you like).

If you want small, then there are plenty of them, and anybody can use
their FAVORITE one. But this is not usually done, it seems that the fact
that the canonical one is “essentially small” is good enough to dismiss
all possible problems.

For example, people which consider the presheaf category
Set^((Set_f)^op)  (object classifier) often do as if Set_f  were
canonical and small.

Now, if you work with a  Grothendieck base topos “as if it were the
category of sets”, you are forced to specify which small category of
finite sets you are using,    or not ?.

Cheers  e.d.

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4 Jul 13:21 2011

### LFCS Free Topos Seminar, Phil Scott, 13 July, Edinburgh, Scotland

If you will be in Scotland on July 13 please consider attending an afternoo=
n of
lectures on the free topos by Philip Scott. You will find the announcement =
below.
-Ben

---------------------------------------------------------------------------=
---

Laboratory for Foundations of Computer Science (LFCS),=20
University of Edinburgh presents:

*****      What is the free topos?  *****

by Philip Scott, Dept. of Mathematics and Statistics, University of Ottawa
SICSA Distinguished Visiting Fellow

http://wcms.inf.ed.ac.uk/lfcs/events/prof.-phil-scott-free-topos-lecture

Location: Informatics Forum, Room 4.31/33, School of Informatics, Universit=
y of Edinburgh, Scotland.
Time: 14:00-17:00, 13 July 2011.

Abstract:

The free topos is a model of intutionistic higher order
logic, and may be thought of as a universe of sets for a moderate
intuitionist.  In this lecture, we give an introduction to free topoi
(on graphs); in particular, we discuss metamathematical properties of


4 Jul 15:11 2011

Concerning Eduardo's question: Now, if you work with a  Grothendieck
base topos “as if it were the category of sets”, you are forced to
specify which small category of finite sets you are using,    or not ?.

No, in the definition of an U-topos, E,  given in SGA 4, Expose IV, § 1,
it is required that E has an U-small (topologically) generating set of
objects.  Bu such a set  or, equivalently, full subcategory of E, is not
part of the data.

Bill Messing

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5 Jul 11:37 2011

Dear Eduardo,

That's right. Specifically, Set_f here has N for its object of objects,
and something more complicated (but geometrically definable) for its object
of morphisms. That way the correct object classifier is defined for any
base topos. (I'm thinking of Grothendieck toposes here, but I guess it
works for any elementary topos with NNO. I even conjecture it does
something useful for arithmetic universes.)

That's a very strong notion of finiteness constructively. It requires not
only Kuratowski finiteness and decidable equality, but even a decidable
total order. Then the category of such finite sets is essentially small,
equivalent to the Set_f I described above. It is the notion of "finite"
needed in finitely presentable algebras, for example in the theorem that
for a finitary algebraic theory T, the T-algebra classifier is
Set^(T-Alg_fp^op), the topos of Set-values functors from the category of
finitely presented algebras. Again, we want T-Alg_fp to be small.

In my paper "Strongly algebraic = SFP (topically)" I was interested in the
situation where, for a geometric theory T, the classifying topos for T is  a
presheaf topos in the form of the topos of Set-valued functors from the
category of finite T-models (and I gave some sufficient conditions for this
to happen). Again, the notion of finite model is this strong notion of
finiteness. However, my main example also involved Kuratowski finite sets,
so the paper discusses in some detail the interplay between the different
notions of finiteness.

Regards,

Steve Vickers.


5 Jul 15:03 2011

### announcement

Advanced Course and Workshop "Large-Cardinal Methods in Homotopy"
will take place in Barcelona, September 1-8. See
http://www.imub.ub.es/hocard11/

Jiri Rosicky

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6 Jul 01:29 2011

### size_question_encore

I have now clarified (to myself at least) that there is no canonical
small category of finite sets, but a plethora of them. The canonical one
is large. With choice, they are all equivalent, without choice not.

When you work with an arbitrary base topos (assume grothendieck) "as if
it were Sets" this may arise problems as they are beautifully
illustrated in Steven Vickers mail.

In Joyal-Tierney galois theory (memoirs AMS 309) page 60, they say S_f
to be the topos of (cardinal) finite sets, which is an "internal
category" since then they take the exponential S^S_f. Now, in between
parenthesis you see the word "cardinal", which seems to indicate to
which category of finite sets (among all the NON equivalent ones) they
are referring to.

Now, it is well known the meaning of "cardinal" of a topos ?.
I imagine there are precise definitions, but I need a reference.

Now, it is often assumed that any small set of generators determine a
small set of generators with finite limits. As before, there is no
canonical small finite limit closure, thus without choice (you have to
choose one limit cone for each finite limit diagram), there is no such a
thing as "the" small finite limit closure.

Working with an arbitrary base topos, small means internal, thus without
choice it is not clear that a set of generators can be enlarged to have
a set of generators with finite limits (not even with a terminal
object). Unless you add to the topos structure (say in the hypothesis of
Giraud's Theorem) the data of canonical finite limits.