Bourbaki and Foundations

[martdowd at aol.com]

FOM:

 Jodmos Horon writes:

So, as far as I see it, their position is more that of a form of agnosticism
that acknowledges that set theory or variants thereof does the job, but also
sort of states that the "intuitive standpoint" is their default position. 
 
 I would like to add some thoughts of my own.
The first task of a rigorous treatment of category theory is to ensure that
mathematical theorems proved using category theory are provable in ZFC.
This was seen as requiring some attention in the case of FLT (Fermat's last
theorem).  Proofs using category theory make use of general facts about
categories, and these are not statements of LST (language of set theory).

A simple solution is to add a predicate C for a particular category, and
axioms stating that C is a category.  Then any theorem provable in this
system,is provable in ZFC, for any definable category for which  the axioms
are provable.  This can be generalized to a finite collection of
categories and functors, and by coding methods to even more general
collections.  It would not be surprising if this method could be used to
formalize the category theory needed for FLT; or for various facts which
caused Grothendieck to resort to universes.  The method applies to functor
categories, provided the index category is small, which is the case in
ZFC arguments.

A second task is to shed some light on the fact that set theory has no
"atomic" elements, which elements of objects are often perceived as.  These
can be added yo set theory (for example Barwise' admissible sets book uses
them).  It is well-known that, although this may be philosophically
appealing and mathematically useful, it is mathematically unnecessary.
Note also that Feferman's "requirement 3" requires a foundational theory to
be able to carry out various set-theoretic constructions.

A third task is to deal with categories that are higher type than a proper
class.  This subject occasionally arises in other contexts, and is readily
handled by adding higher types to ZFC.  In his 1964 book, Paul Cohen
mentions the system that has types of every finite order.  Note that any
theorem of this theory can be interpreted in any $V_{\kappa+\omega}$ where
$\kappa$ is a strongly inaccessible cardinal, and is a true statement.  It
is reasonable to hold that these statements are true statements about a
hierarchy of "meta-objects" erected atop the cumulative hierarchy.  My
paper
 Higher type categories
 Mathematical logic Quarterly 39 (1993), 251--254
proves some facts about some such categories.
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