[FOM] Foundations of Mathematics: Set Theory vs. Category Theory

[Oliver Wienand]

Hello

I'm a student at the University of Kaiserslautern, Germany. I plan to give a
seminar talk on FOM. In particular I want to compare set theory based
foundations with category theory based ones.
I want to ask you, if you have good ideas or literatur which I should take
in.
To the background of me and the other participants:
We know predicate logic level 1 (this is 99% not correct in english, I hope
you understand), the ZFC and NBG axiom system, the completeness,
incompleteness and indepence results by Goedel and Cohen.
Then some alternative axiom to AC and GC:, Freiling's axiom of Symmetrie,
the Axiom of Determinateness and V=L.
Also we know a little bit category theory, maybe up to the definition of a
topos and the topos SET in McLarty: "Elem. Categories, Elem. Toposes".
But there is generally no background to measure theory, number theory or
graph theory.

Until now I will mainly take three (four) parts:

1. Why do we need Foundations and how many?
2. Category theory based on set theory.
3. Set Theory on Category Theory
(4.) Further use of Category Theory in Math and Computer Science

For Part 1 I have read many messages in the FOM - Archive and
Comments/Introductions to set theory or category theory. But I'm a bit weary
about that.
At the moment I think fom should provide

a) the basis mathematical discourse. This means a language, in which we can
talk to each other without to many missunderstandings. I think this part ist
many covered by the predicate logic.

b) a framework in within you should be able to do most of mathematics. So
that you could interlink totally different parts of mathematik easy, because
they are build of the same stuff. Here I think ZFC is the "standard". Which
I find very funny. Because if ZFC is consistent, then it has an countable
modell (Löwenheim Skolem). And this fact is not really in line with math.
common sense.
To the framework notation I would also cite Colin McLarty in "Elementary
Categories, Elementary Toposes" where he describes that the need for a
foundation arises, when you exaim multiply structures.
He says that the axiom of a category (and here I think you could fill in
every structure) can stand for it alone, but if you try to build structure
out of two categories, or similiar things, you need a background, where your
categories do live in.

c) tools, which gave you the power to take the fom under the microscope. So
you could say thing like the incompleteness theorem or similiar.

Now, how many fom's do we need? When I looked at it for the first time, I
would have said one, but a good one. But now I would prefer a multiple
system of FOM's and studies how they are related to each other. This may be
the state at the moment.

For part 2 I have a lot of material, the Grothendiek universes, inaccessable
cardinals, etc. on the bases of ZFC or NBG. Then I've found an article by
F.A. Muller "Sets, Classes and Categories". Here he describes the
ACKERMANN - set theroy and a slight extensions to give a basis for category
theory.

But my hardest time I have with part 3. At many papers there is a hint to
category foundations, but I have not found a work, which describes this
categorical foundation. Maybe I just didn't see it, when it was under my
eyes. So I hope you could give me a hint here.

I hope you could give me some ideas.

sincerely

oliver wienand