FOM: Re: Categorical foundations

[Martin Schlottmann]

The discussion on this topic has become a little
bit loaded, and I think that arguing about the question
of which system, category or set theory, is more convenient
or simple or basic or intuitive or whatever will not
lead to any productive result. Instead, I would propose
to embark on the following program to unify both approaches.

In order to explain what I would like to see let me
give the following (too?) simple example which one could call
"Set vs Class Theory". Consider the following two
systems: ZFC set theory and GNB class theory. Both can
be used for foundational purposes, and it is very
easy to translate between both languages: ZFC is a
sub-system of GNB and, on the other hand, GNB is a
conservative extension of ZFC. Therefore, every
mathematical reasoning in the framework of one system
can be immediatly transformed into the other system
and vice versa. As a consequence, it would be futile
to argue if classes in the sense of GNB are admissable
for f.o.m. purposes or not. It would also be futile
to argue a priori which system is more convenient;
if one likes, one can use both systems almost simultaneously
without having much to worry about contradictions.
If one system is superior to the other, this will simply
sort out by itself in due course; if none is really
better, it doesn't matter if both will be around indefinitly.

Translated into the category vs set discussion:
Let both, proponents of sets and proponents of
categories, fix one system each which they think suits
their needs for formalizing mathematical reasoning.
Say, the system of set theory is ZFC, the system for
category theory is XYZ (I apologize for being not familiar
enough with category theory to make a definite proposal
here). Then, set theorists translate XYZ into ZFC,
categorists translate ZFC into XYZ. After this, write
a textbook and simply let mathematicians do their work
in whatever of the two systems they like and see which
system survives.
If both systems stand the test in everyday life, then,
obviously, none of them is more convenient for f.o.m.

If it turns out that one of the systems cannot be
translated into the other, then there is a serious
discrepancy on the admissable principles of mathematical
reasoning. E.g., if it turns out that XYZ needs ZFC+
inaccessible cardinals, then one can argue seriously about
the question if XYZ uses principles of reasoning which
are not generally accepted (for myself, I would not
like to have my own results to depend on the existence
of inaccessible cardinals). At this stage, one has
a point which can be productively discussed.

I apologize in advance if the program I described has
already been carried out and only escaped my attention.
If so, I wonder what all this discussion is about, after all.

I also apologize being elaborate up to the edge of
triviality, but I wanted to make myself clear in an
already loaded discussion.


-- 
Martin Schlottmann <martin_schlottmann at math.ualberta.ca>

Department of Mathematical Sciences, CAB 583
University of Alberta, Edmonton AB T6G 2G1, Canada