FOM: Re:The missing 1%<

[Andrej.Bauer@cs.cmu.edu]

"Hasan Keler" <nx at cheerful.com> writes:
> >I should think that category theory is the remainder.
> 
> To be honest, I don't know much category theory. Are categories proper
> classes in general rather than sets? Is this your reason why you don't
> consider category theory as something not in the reach of ZFC? What if we
> use NBG?

In general, categories are proper classes (the category of all groups,
the category of all topological spaces, the category of all sets).
Many important examples of categories are actually just sets (the
"small" categories). But this is not a reason to consider category
theory as something in the reach of ZFC.

> Naturally, the following question comes to me:
> 
> Is there a theorem of "ordinary" mathematics that can NOT be
> translated into the language of set theory ?

All of category theory can be translated into the language of theory
of classes and sets. But that is an extremely misleading answer,
because it suggests that any foundational questions in the realm of
category theory are automatically reduced to questions about class/set 
theory.

Let my try to explain why, in a sense, category theory is not "within
the reach of set theory", from the foundational point of view. It
should then follow that "the missing 1%" is quite a bit more than 1%.

There is more to foundations of mathematics than just questions about
consistency, proof strength, independence results, and other such
popular matters that a logician, set theorist, or a model theorist
might consider.

A goal of foundations of mathematics is to expose, clarify, and study
those mathematical concepts and ideas that are fundamental for all of
mathematics, or at least very large portions of it. The concepts such
as "set", "class", "function" come to mind, and also more basic ones,
such as "language", "proposition", and "proof". These concepts are
indispensable in ordinary mathematics.

To a degree, what is and isn't a fundamental concept is a matter of
opinion. For example, is "cardinal number" a fundamental concept?

There are concepts that are fundamental but are not at all exposed,
clarified, or easily studied within set theory. These are the basic
category-theoretic concepts, such as "natural transformation",
"adjoint functor", "limit", and "colimit". For example, the notion of
an adjoint functor is fundamental to algebra, geometry and logic at
the same time, and it brings to light deep connections between these
branches of mathematics. It is without a doubt of interest to
foundations of mathematics to study such concepts, in order to see
what else is out there apart from sets, sets, and still more sets.

Category theory and classical set theory are two different points of
view. Each has its own advantages. They are both translatable into
each other, but that is not the end of the story. Category theory is a
foundation of mathematics in its own right, not because it offers
things that cannot be expressed in set theory, but because it makes it
possible to study fundamental concepts that are invisible from the
set-theoretic point of view.

You should have asked instead:

``Naturally, the following question comes to me:

 Is there a theorem of "ordinary" mathematics that can NOT be
 NATURALLY translated into the language of set theory?''
 ^^^^^^^^^

Everything in ordinary math CAN be expressed in set theory.
And everything in ordinary math CAN be expressed in category theory.

Without set theory we would never think of studying connections
between consistency, large cardinals, and combinatorial statements.

Without category theory we would never discover adjoint functors, and
so we would never know that such things as the construction of a free
group, the completion of a space, and logical quantifiers, are all
examples of a single concept.

Thus, category theory is not part of set theory, and vice versa. And
there you have your missing 1% and more.

--
Andrej Bauer
Graduate student in Pure and Applied Logic
School of Computer Science
Carnegie Mellon University
http://andrej.com