Bourbaki and Foundations

[jodmos.horon]

Hi.

> Harvey Friedman: I want to bring in some facts that might better direct this discussion.
> The first is from the very famous and very influential book
> [1] Saunders Mac Lane, Categories for the Working Mathematician
> I quote from page 10:
> A category (as distinguished from a metacategory) will mean any interpretation of the category axioms within set theory. [...]
> So it is clear that Mac Lane explicitly uses set theory as the foundation for category theory without hesitation, right from the beginning.

Well, in fact this observation goes back even to the original paper of MacLane
by MacLane and Eilenberg in 1945. They did not mention set theory explicitly in
that paper, as far as I recall. But, clearly, it's taken to be a model of some
axioms.

But, on the other hand, so is set theory: it's all about a model of ZFC...

So, whether one chooses set theory or the notion of model as primitive, it's a
chicken and egg question. Of course, you do have an axiomatic setup for set
theory that allows one to create models and categories as such models within
it. On the other hand, one can take a model of set theory, and carve out from
it all the mappings between sets. And one then gets a model of the category of
sets from the model of ZFC. Or of the category of binary relations (I like that
one better...)

And one can then do model theory on that category of sets carved out from a
model of ZFC itself seen as a model. Pretty much the same way as we do
considering models of ZFC.

>From where I stand, what I'd like is not set theory, but an axiomatisation of
models. A bunch of ZFC-like axioms that would allow us to axiomatise models the
same way ZFC axiomatised sets.

And it may perhaps be possible to do that in a structuralist feel that would
take lessons from what category theory did teach us in terms of naturality and
universal problems.

> How can this be explained?
> Well I think that Mac Lane realized that any attempt to remove set theory from the picture, and have notions like category as primitive, is likely doomed not only in terms of its complexity, but also in terms of its Philosophical Coherence.

The only objection I have is that one cannot avoid attempting to justify on
philosophical grounds that sets exist and that there is a model of it somehow
outside in nature. There needs to be philosophical arguments (not formalist
ones, we know they failed) to claim that sets exist in some way or another.

The same goes for a notion of a model. In fact, I believe that it's more or
less an equivalent question.

So this does not preclude considering a category (of sets) itself as a model at
the same ontological level as one would put models of set theory.

> Or you can adopt Mac Lane's approach in [1] and have a page or two with precise definitions to start things out sitting on top of set theory.

Well, I'm still not satisfied with the axioms MacLane and Eilenberg gave. There
are hidden subtleties about associativity that arise even in the special case
of groups that people do not seem to care much about, though I do find them
rather curious to study in themselves.

> For getting the real mathematics going in the sense of mathematical culture, it doesn't make any difference which of these two. If philosophical coherence is desired or at issue, the second way is best. At least until there is a philosophically coherent version without sitting on set theory.

What would make set theory more "philosophically coherent" than an
axiomatisation of constructions in model theory itself ?

> The mutual interpretability of set theory and category theory of various kinds is very strong in that they preserve the truth of statements about the ring of integers and even the ordered field of real numbers with distinguished part Z.
> So basically, the point is that for reasonably well grounded mathematics of a standard kind, the mathematics one has is completely interchangeable - and automatically.

What kind of mutual interpretability of set theory and category do you have in
mind, precisely ? I'm not really convinced that any formalisation of category
theory or of a specialisation thereof has managed to adequately interpret ZFC.
I'd be curious to know. I know quite a lot of mimickry can be done, but I've
yet to see a proof of interpretability. But perhaps I haven't looked far enough.

Best regards.

Jodmos Horon.