FOM: Picturing categorical set theory, reply to Silver

Charles Silver csilver at sophia.smith.edu
Wed Jan 21 15:06:36 EST 1998

My basic question, to which Colin McLarty replied was how
'Function_C' (the formal notion of "function" in category theory) relates
to 'function' (the informal notion of "function" in ordinary English).

I said:
> >	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.

Colin McLarty said:

>         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.

[lots SNIPPED]

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

I think so.  But, I'm looking for something slightly different.
Perhaps what I'm looking for isn't there.  The kind of thing I'm looking
for is an underlying *conception* of function that is *explicated* by
category theory.  (Maybe "explication" is too strong here.) There are a
couple of examples I know of for set theory. Around 1968, George Boolos
wrote a paper, called "The Iterative Conception of Set." (I think that was
the title. I don't have the paper before me.)  In that paper, Boolos laid
out an underlying *conception* of sets.  Then, he showed that all the
axioms of Zermelo set theory were true for the conception he spelled out.
That is, given the underlying conception he presented, much of set theory
followed.  (As I recall, Boolos didn't think AC followed from the
conception itself, but that is was "an attractive further thought.")

Also, Shoenfield in his book does something quite similar.
(Starting on p. 238) He presents an "intuitive notion of a set."  Based on
this intuitive notion, Shoenfield shows that "the Russell paradox
disappears."  He also explains his "principle of cofinality" for this
conception of sets.  Then, once he's through explaining the concept he has
in mind, all of set theory follows.  That is, the axioms of set theory are
*true* for this conception.  For example, Regularity follows, as does
Replacement.  I have some qualms about Shoenfield's conception (for
example, he relies on a kind of "visualization" principle that seems vague
to me), but what Shoenfield and Boolos do are examples of the kind of

I'm asking whether there is some "intuitive notion of function"
that can be used as a basis of 'Function_C' (i.e., `function' in category
theory) that would be somewhat like Boolos's iterative conception or
Shoenfield's "intuitive notion of a set."  That is, is there an underlying
*conception* of function that can be spelled out, so that the axioms about
functions in category theory are *true* for that conception?

Charlie Silver
Smith College