FOM: Picturing categorical set theory, reply to Silver

Colin McLarty cxm7 at po.cwru.edu
Wed Jan 21 12:20:59 EST 1998

```Charles Silver wrote:

>
>	I want to raise some elementary concerns about the picture of
>category theory under consideration.  First, I'll present a very basic,
>perhaps very naive picture of set theory:
>
>	Almost all beginning set theory textbooks get off the ground by
>stating that the central concept is that of a "collection," an
>"assemblage," an "ensemble," etc.  On these accounts, the beginning notion
>is one that we are fairly familiar with in real life, via counting and so
>on.

Yes, and up to this point there is no difference from categorical
set theory. Finite sets of concrete individuals (such as playing cards on a
table) look the same in ZF with urelements as in categorical set theory.

Now look at infinite abstract sets, or "pure sets". ZF gets these by
transfinite type accumulation. Category theory gets them by considering
functions.

The picture is: What matters in set theory is not "what" elements a
set has but "how many". We tell how many elements a set has primarily by
seeing what functions it has to and from other sets. For example the set of
integers has a one-to-one function into the reals, but there is no
one-to-one function the other way, so there are "more" reals than integers.

A function from a set M onto a set S is intuitively a kind of rule
by which each element of S is bound to one or more elements of M, but no
element of M is bound to more than one element of S. But this is only
intuition. To be precise, we do not define "function" at all but develop
axioms saying what functions exist. (Of course since Goedel we know these
axioms can never be complete for set theory including arithmetic).

We describe the cartesian product AxB of sets A and B by saying
it has as many elements as there are pairs <a,b> of an element of A
and an element of B. But we do not say what its elements "are". We say an
ordinal number has as many elements as predecessors, but we do not say what
its elements "are". And so on.

We could develop this farther, but I'll stop and ask whether this
picture makes any sense to you? Can you see it?

Colin

```