Dividing by Zero and Axiomatic Set Theory

From: John Savard (jsavard_at_excxn.aNOSPAMb.cdn.invalid)
Date: 12/23/04 

Date: Thu, 23 Dec 2004 07:48:49 GMT

One of the major questions in the philosophy of mathematics is whether
or not mathematical objects are "real", that is, having an independent
pre-existence, or whether they are human creations.

A recent thread, begun by someone asking if zero was even or odd, led to
a discussion of division by zero.

I noted that one could indeed say that 0/0 equals just about anything,
and a finite number divided by zero equals both positive and negative
infinity - but doing so is generally viewed by mathematicians as
pointless and unproductive. Thus, mathematicians concern themselves with
the real numbers.

This seems to relate to the philosophical question above in this way:
one can perhaps think of a wild and wooly reality of "quantity" or
"number" which is pre-existing, and which is manifest in many physical
processes in nature - and then one can also think of a subset of that
reality, abstracted from it, which is chosen by mathematicians to serve
their purposes in efficiently finding information which is applicable
both to the formal system of arithmetic over the real numbers, and to
the original wild and wooly reality from which it was abstracted.

I don't think it settles the philosophical question here, though,
because the real numbers are a very natural choice of abstraction from
the fundamental natural notion of quantity.

As we know, there are many small villages in the real world.

In some of these villages, there is only one barber.

In some of those villages, the barber is an adult male.

In some of those villages, every adult male in the village is
clean-shaven.

And in some of those villages, the adult males are creatures of habit in
how they arrange to be shaved; some will always shave themselves with a
safety razor at home, looking in their bathroom mirror, and others will
go to the barber shop, and sit down in the barber's chair, and be shaved
by the barber who stands behind him.

In such a village, of course, the barber, since he cannot both be
sitting in the barber's chair and standing behind it at the same time,
must instead shave himself at home, before his bathroom mirror.

Thus, Bertrand Russel's barber paradox is due to a slight error in
phrasing: what one should say is that every man in the village either
shaves himself, or is shaved by the barber _in his capacity as barber_.

But that doesn't change the fact that there definitely is a paradox when
one talks about "the set of all sets that are not members of
themselves".

Here, again, we begin with a wild-and-wooly reality; the world of sets
that naive set theory attempts to describe.

There are objects; these can be numbers, physical objects, or other
abstractions such as names, words, or dates. There are sets; a set has
the property that for every object, the question "is this object a
member of that set" has a yes or no answer. Sets are objects.

Clearly, here, the set of all objects, and the smaller set of all sets,
are perfectly meaningful.

But while the statement "Set X is a member of itself" is in itself
meaningful and non-paradoxical, it still can't be used to define a set.
While it is nice to know that each set leads to the statement "Object A
is a member of set S", it would be very useful to have a rule for going
from statements to sets. Obviously, the statement "x is a real number,
and it is less than 3" leads to the set of real numbers less than 3; but
if the statement "X is a set that does not contain itself" does not lead
to a set, we need to know why.

One possible rule that might be tried is "a statement which is either
true or false of all objects leads directly to a corresponding set of
objects for which that statement is true if the truth of that statement
for an object is not affected by that object's membership in the set
corresponding to that statement" to eliminate self-reference.

Note that the conditional is "if", not "if and only if".

But that implies we _can_ have "the set of all _other_ sets that do not
contain themselves". This set doesn't contain itself.

But then, can we call that set Joe, and have another set Fred that is
"the set of all _other_ sets that do not contain themselves", which
doesn't contain Fred, but which does contain Joe?

Or, what about "the set consisting of 'the set of all _other_ sets that
do not contain themselves' and all other sets that do not contain
themselves, except this set"?

To avoid these pitfalls, a well-behaved portion of naive set theory was
abstracted out, known as axiomatic set theory, based on the
Zermelo-Frankel axioms, with or without the axiom of choice.

So, if one takes the position that there is a pre-existing wild and
wooly reality, and the attempt to describe it by naive set theory
failed, from which a part called axiomatic set theory was extracted...

then one can meaningfully ask whether or not the axiom of choice, or the
Continuum Hypothesis, or the Generalized Continuum Hypothesis, is true.

What that would mean is that, if one tries to bite off a *larger* chunk
of the wild-and-wooly reality people have in mind when they talk about
objects and the sets that contain them, and one succeeds in biting of
such a chunk in which these questions are no longer undecidable - how
will they be decided?

One can extend axiomatic set theory _either_ way for any of the
questions undecidable within it, but the "right" answer to those
questions is the one consistent with that vague thing which naive set
theory attempted to describe.

Unlike the real numbers, which are a highly satisfactory abstraction
from "quantity", leaving us with no real motivation for extending the
real numbers to allow division by zero, axiomatic set theory is not a
large enough chunk of naive set theory to be fully satisfying.

Because a wild and wooly reality is not tamed with axioms and
postulates, nothing can be directly deduced about it - but the wild and
wooly realities are still real. They motivate our choice of which formal
systems to talk about. Physical reality led us to talk about the
properties of numbers, and about the properties of sets.

Thus, "is the continuum hypothesis true" means "will the physical
reality of objects and collections motivate us to extend axiomatic set
theory to include postulates leading to the continuum hypothesis being
true".

John Savard
http://home.ecn.ab.ca/~jsavard/index.html

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