Re: A quiet query from a visitor

On Wed, 22 Aug 2007 07:30:42 -0500, David C. Ullrich
<ullrich@xxxxxxxxxxxxxxxx> wrote:

On Tue, 21 Aug 2007 11:54:06 +0100, Angus Rodgers
<twirlip@xxxxxxxxxxx> wrote:

[...]

(The closest I come to agreeing with you is probably that
I cannot come up with an interpretation of set theory. I
really don't know what sets are. Also, I don't think that
regarding "set" as an undefined term, like "point" in some
axiomatisation of Euclidean geometry, solves the problem,

It's a simple fact that there _must_ be at least
_one_ undefined term - whatever body of definitions
we decide to adopt there will be a _first_ definition
on the list, and the terms used in the first definition
will be undefined. Why do you object to "set" and
"element of" being undefined, and what would you
_prefer_ we use?

Seems to me the point is that as long as we need to
have some undefined terms, "set" and "element of"
are a good choice, because we _can_ then define
more or less everything else we want using just
those two. You have a better proposal? Or do you
want a system with _no_ undefined terms or what?

I would like to know how one can talk of models of
theories at all, without appealing to some naive
notion of set or class. But I tried discussing this
in sci.math ages ago (in my very first post, I seem
to recall, in or around 1991). I got nowhere then,
and I expect to get nowhere now. But surely at least
my puzzlement is understandable (and shared at some
level, however stupid that level might seem to be).

Suppose you somehow define a formal theory (without using
some sort of naive set theory to define it). Then you
want to talk of "models" of said theory. But a model is
a set or class or collection of elements (or an ordered
collection of collections of elements of different sorts,
together with various functions defining interpretations).

So model theory is a part of set-theoretic mathematics
already. So you cannot appeal to models of theories
containing an undefined term "set" in order to explain
anything about the "set theory" you need to use to do
model theory. And surely not only to do model theory,
but also to do everyday mathematics: for what is a
group except a model of the theory of a group, and
what is a category except a model of the theory of a
category? (But see next paragraph but one.)

Surely "set theory" is not a particular theory within
mathematics (like Euclidean geometry, or group theory),
but an aspect of the axiomatic method itself? Not that
this in any way rules out /treating/ set theory as a
purely mathematical theory, like any other, and learning
extremely valuable things about it that way. It's just
that it cannot tell you anything about the naive set
theory you are already using all the time, including
when studying formal theories of "sets" (in the sense of
an undefined term satisfying certain formal postulates)
in a sophisticated way. That would be rather like
thinking because you know how to study some kind of
mathematical "space" in general, you no longer need any
physical space in which to store your notes, or sit at
your computer.

There is a possible converse kind of argument to this,
which is that although a category or topos can be treated
just as a model of a mathematical theory like any other,
this does not imply that category theory or topos theory
does not possess some kind of fundamental significance
in the same way that set theory does. But I haven't a
clue what that might be.

What I would be happier thinking about, but am not yet
in a position to think about (because I just haven't done
enough mathematics), is how well our formal theories of
sets match up to the informal kind of set theory that it
seems natural (even to me) to use in everyday mathematical
practice. My intention is to observe my own work as I
continue trying to learn mathematics, and reflect on why
I believe that what I am doing makes sense. I gave one
example of this the other day: when doing a beginner's
exercise in Galois theory, it seemed natural to me to use
a Cartesian product of two symmetric groups, even though
I have not the slightest idea what a Cartesian product
is supposed to be. (I know vaguely that a set is supposed
to be a kind of collection, and unions and intersections
make fairly good sense, but ordered pairs make no sense
to me at all. On the other hand I appreciate the great
technical advantage of using them. If I remember right,
Whitehead and Russell had to repeat essentially the same
arguments for relations as for classes, because this
marvellously neat technical trick wasn't available to
them.) This seems like a creative internal tension. One
part of my mind is doing maths in an ordinary way, while
another part of my mind sits outside and asks annoying
and possibly cranky or just ill-informed questions. I
would far rather have this kind of internal debate with
myself than get embroiled, on one side or another, in
long threads about foundational matters in sci.math! But
for the moment I seem to be stuck with having to say what
I think, even though I know that I don't yet know what to
think.

I'm sorry this isn't clearer. But it's a confusion, one
of several, that goes back to around 1971, so I don't
expect it to get sorted out quickly. And I really don't
mean to go on about it! It's just that when a relevant
topic comes up in sci.math, it's very hard to know whether
to just keep quiet or not. Neither choice seems right.
I'm bound to say too little or too much - in fact, too
little (because after 36 years of intermittent brooding,
there's much more I could dredge up) AND too much (as
my posts in these threads always seem to be too long).
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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