Rethinking Set Theory

Rethinking Set Theory:

Over the last few years, I’ve been very slowly working up a short expository
paper — requiring no knowledge of categories — on set theory done categorically. It’s now progressed to the stage
where I’d like to get some feedback.
Here’s
the latest draft.
Typos, clumsy wording, mathematical matters: I want to hear
it all.

I have one request, though. If you do leave a comment, please take more
time than you usually would to make sure it’s (1) carefully worded, (2) respectful
to other people, and (3) scrupulously polite. Sorry to ask this: normally I
wouldn’t feel the need, but there’s an unfortunate history of discussions
of categorical set theory turning bad-tempered, and I really want to avoid
that happening here.

So, if you’re composing a comment and you feel yourself getting hot under
the collar, please save a draft, sleep on it, and come back later when your
temperature has returned to normal. There’s no hurry: this blog isn’t
going anywhere.

I’d also like to hear about anything I’ve written in my draft that
you think is overstated. (I’m particularly keen to
hear about this from people who fundamentally share my views.) This
paper is intended to be thought-provoking, and I know there are
parts with which some people will disagree. However, it’s definitely not
supposed to be inflammatory. I want every statement I’ve made to be careful
and measured, so I’ll be grateful if you can help me find places where I
might have slipped up.

I won’t write much here about the
paper,
since it is itself expository.
Experts will immediately recognize it as a description of Lawvere’s
Elementary Theory of the Category of Sets, which states that sets and
functions form a well-pointed topos with natural numbers and choice.
(Semi-experts might want the help of
this supplement: in my draft, I’ve
slightly rephrased the standard axioms, but the supplement proves that the rephrasing makes no difference.)

However, it’s very much not written for experts, or even semi-experts. No
knowledge of categories whatsoever is needed to read it. Nor do I attempt
to teach the reader anything about categories.

What’s this paper supposed to achieve? I’ll quote from the first page:

… many of us will go our whole lives without learning ‘the’ axioms
for sets, with no harm to the accuracy of our work. This suggests that we
all carry around with us, more or less subconsciously, a reliable body of
operating principles that we use when manipulating sets.

What if we were to write down some of these principles and
adopt them as our axioms for sets? The message of this article is
that this can be done, in a simple, practical way.

There are probably many ways of doing this, but the one I describe is categorical in spirit.

Relatively few mathematicians have heard of categorical set theory, but my
experience is that among those who have heard of it, there are some common
misconceptions. These misconceptions are largely a result of bad
communication, so I’ve done my best to address them directly.

One misconception is that because of the involvement of categories (and
even toposes, God forbid), categorical set theory must be awfully
sophisticated. I try to dissolve that assumption by stating the axioms in
a wholly elementary way, not using the word ‘category’ once. The box
on the first page shows that the axioms—informally paraphrased—are just
a bunch of completely mundane statements about sets:

1. 
   Composition of functions is associative and has identities.
   


2. 
   There is a set with exactly one element.
   


3. 
   There is a set with no elements.
   


4. 
   A function is determined by its effect on elements.
   


5. 
   Given sets X and Y, one can form their cartesian product X×Y.
   


6. 
   Given sets X and Y, one can form the set of functions from X to Y.
   


7. 
   Given f:X→Y and y∈Y, one can form the inverse image
   f −1(y).
   


8. 
   The subsets of a set X correspond to the functions from X to {0,1}.
   


9. 
   The natural numbers form a set.
   


10. 
    Every surjection has a right inverse.
    

As I said, this is an informal summary, and there is some distance
between the statements above and the precise versions (which you can
find in the paper). However, if I’ve done my job properly, even the
precise versions should come across as run-of-the-mill, unremarkable, statements about
sets as used in everyday mathematics.

I also address the misconception that, because the definition of category
uses something like the notion of set, categorical set theory must be
circular. But I won’t go into that here, because you can find it in
my
draft —
which I’m hoping you’re about to go and read, then carefully comment on here.

DIGITAL JUICE