FOM: Theorizing About Theories I: Cantor's subsets and the beauty of theories

[Dennis E. Hamilton]

[This is a re-identified resend of my first note on this topic.  It
apparently vanished into the ether sometime last Thursday night.  There is
resolution to (1), below, in the already-distributed part II.]

I have been mulling this over while doing other things, then I forgot so
much of it I had to start over.  I want to see if I can find a way to
distinguish what is involved here.

I want to look at three things: (1) The fundamental notion of subsets and
set identity, (2) the relationship of the formalization of sets to how one
establishes that there is a subset relationship without somehow
materializing the sets and parading their members, and (3) whatever that
might show us about the solidity of the theory despite how we can easily
misread an interpretation back onto the formalized theory.

I am engaged in applications of theories and there it has been easy for me
to go astray.  So I want very much to have an useful explanation that
reminds me of the perspective to take as well.  Also, I am offering this as
a perspective that is valuable, not like it is some absolute truth.

1.	CLEANING UP SUBSETS AND IDENTITY OF SETS

In a related note, I observed that

1.1	A is-a-subset-of B is equivalent to every member of A is a member of B.

1.2   And that there is a nice harmony in having

	(A is-identitical-to B) is-equivalent-to (A is-a-subset-of B) and (B
is-a-subset-of A)

1.3   I failed to notice that 1.2 can be an *axiom* for the identity of two
sets.  Now I can't find where I ran across that.  It basically has the
identity of sets be based on having the same members, which is always the
principle.  This principle has nothing to do with the cardinality of the
sets involved.  It applies for all sets.

2.	HOW ONE ESTABLISHES SET IDENTITY AND SET EQUIVALENCE

If sets are finite and they can, in principle, be written down, one can
imagine carrying out a parade-ground process of matching up the "identical
twins" and seeing that there is no one without a twin in attendance.

But that's an experiment in an interpretation of (finite) sets, and it isn't
about what the formal theory requires.  It is also painful to do in an
applied set theory, since the available deductions are so weak and the
details that have to be axiomatized so numerous.  So we might use a
computer, and indeed we do in practical situations.  Database systems earn
their keep by rapidly synthesizing correspondences among (formal) finite
sets.

But to use the deductive system, we want to do a deduction, not a
construction involving exhaustive enumeration.   Then we are operating in
the theory (or the metatheory) and we not depending on the particular
interpretation that we might have in mind (other than we are talking about
sets of non-negative integers), provided that it is a valid one.

Step 1.  So, for example, one could show that the non-negative even numbers
are a subset of the set of non-negative integers simply by some deduction
based on the definition of the two sets.  [I am intentionally not providing
it here, I want to see if the explanation holds up by itself.  This will be,
as it were, a formal demonstration to be carried out.  Notice all of the
agreements we make, in language, on accepting the definitions of the two
sets.  Let me know if this works.]

Step 2. Then one shows that the sets are not *the* *same* by demonstrating a
member of the set on non-negative integers that is not an even number, so it
is not in the subset.

Step 3. How do we show that the set of even numbers is of the same
cardinality as the set of all the non-negative integers?  By deduction based
on a formal correspondence, not an actual parading.  For example, show that
every non-negative natural number identifies a member of the subset and that
nothing in the subset is left out in so doing.  Why do we do it that way?
because that is what the formal theory establishes as "having the same
cardinality" or "being equivalent (in terms of power), etc.  It is made up.
And we keep it because it works great, it is consistent with the other rules
we might have, etc.

This is almost too easy in the case of the set of even non-negative
integers.  The temptation, for me, is to have it seem so simple that I think
something is missing.  Or I will try to interpret it mid-way.  Bad idea.  Do
the deductions.

OK, you can do it with the set of odd numbers, and then for one more show
that the set of even numbers and the set of odd numbers are also of the same
cardinality.  (No, don't stop and say what is happening is that the numbers
have been evenly divided up. That interpretation would lead you to things
like supposing that there are an even number of numbers or something like
that.  Just stay in the formal game.)

The important thing to notice about what the formal theory provides, here,
is that we operate by deduction from a set of axioms that we are pretty
confident about.  At the end of the deduction, we can look at an
interpretation and how it follows there too.  And the interpretation might
involve only something that can be carried out in principle.  That is to
say, the formal theory has gone beyond what I can confirm by experiment or
even pencil-and-paper construction.  But no matter how far I carry this out,
I reach no contradiction, and the in-principle outcome remains.  It might
also take me to a place where I can conceive of no interpretation.  Consider
it a bonus, not a liability.  Nothing of particular use but an example of
the beautiful reach of the theory.

One more thing.  The confidence we have has to do with how interpretations
we care about, and how the "natural" ones (operating with numbers all some
reasonable proximity to 0), all work out simply and directly.  That the
theory takes us beyond what is "natural" or available to the intuition is
one of the wonders of theorizing.

And if you don't need the transfinite for application to an important
interpretation, you can certainly do without it.  I don't need it to
reconcile my bank statement, and I won't miss it.  But some theories are
more powerful and easier to apply if we don't make an effort to limit their
application.  The application will likely be limited, but the theory will
not be limited to that application.  That's one of the beauties of
arithmetic.  It is one of the beauties of set theory, even if I will never
ever need to deal with a really "large" cardinality in anything that
interests me in my personal life or my life as a scientist and a scholar.
Maybe not even very many not-so large cardinalities beyond the cardinality
of the natural numbers.

3.  THEORIZING ABOUT THEORIZING

I just gave you my personal theory about what the application of formal
deductive theories is like and how valuable it is.  If I could motivate this
with a picture I would.  (I am looking for that in my pet project, which is
about how we manifest theories in the behavior directed by computer
programs.)

What I want to do is forego the discussion about how we arrive at theories
and how we refine and formalize over time.  Even how we acquire mathematics
in our development as individual human beings.

I want to start from the vantage point of there already being theories and
we are equipped to apply them, and we have some mastery of them, like having
mastery of more language (like Italian as well as English, or reading,
writing and performing music).  We learned the theories, however we do, as
such, using whatever ladder we climbed to do it.  Now look from the vantage
point of having mastery of a theory.

First, consider that it is valuable to consider that a theory is not talking
about anything at all.  That is, no thing in particular.  Yet it deals with
some kind of structure or pattern for which there are rules and principles
embodied in the theory.  And there may be useful interpretations.  The
beauty of some theories is certainly in the marvelous power that we have
when the interpretations are valuable for practical purposes.

The interpretation of s = vt as a condition of distance covered at a fixed
velocity over a given time works beautifully.  All kinds of observations in
the natural world become predictable by inference from interpretation of
such theories (and most of the inferences are accomplished by convenient use
of arithmetic).  No matter what a Zeno would say about how motion works, I
can count on the interpretation taken from s = vt working every time (to
within the usual allowances for imperfect conditions).  This, by the way is
perhaps a good illustration that theories, necessarily abstracted, never
explain.  Valid interpretation gives us what, but not how.  And certainly
not why.  No matter how much that is what we want to know.  The only "why"
that a deductive theory will answer is one that is the consequence of a
deduction.  It's not a "why" for the situated world of our experience.

That's probably more than is useful to say at this point.

Does this provide any value in struggling with questions about the
cardinality of transfinite sets, and formal deductive systems generally?  I
want it to.  It is at the foundation of what I am trying to work out about
the application of computers, and I would like to offer a powerful way to
view the connection between (deductive) theories and the subjects we apply
them to.  I would love to find where this is said better and I can move on
to other problems.

-- Dennis

[PS: It is Mr. Hamilton, though I have friends who call me Professor
Hamilton.  I take it as a complement.]

------------------
Dennis E. Hamilton
http://orcmid.com/
mailto:dennis.hamilton at acm.org
tel. +1-206-932-6970
cell +1-206-779-9430
     The Miser Project: http://miser-theory.info


-----Original Message-----
From: owner-fom at math.psu.edu [mailto:owner-fom at math.psu.edu]On Behalf Of
Everdell at aol.com
Sent: Sunday, June 23, 2002 13:51
To: fom at math.psu.edu
Subject: FOM: RE: Cantor's subsets

[ ... ]

True.  I struggle with the notion of set as a container--possibly a bag open
at the top to allow for infinite stacking.  However, after much, and quite
sympathetic thought, my question seems to me still open.  Now that a subset
is defined in English in terms of the preposition "in," as in(?) "(A
is-a-subset-of B) and (there exists an x in B for which x is not in A)."  I
have to ask what "in" means--especially if x is "in" an A or B that has no
"bounds," and if one-to-one correspondance is what is not to be assumed but
proved.  This will probably take us back to the definition of set-element,
and here, too, it seems to me, some of the difficulties posed by infinite
sets have been avoided or elided rather than confronted.

Take Cantor, for example (and my thanks to Dennis Hamilton for reading the
source we both have before us more carefully than I seem to have done).

[ ... ]

Still, it seems to me that with Cantor we are back again to the difficulty
of
explaining what the set-elements or "members" of an infinite set are, since
a
set B, which is a "part" or "Besandteil" of another set A, has no elements
which are not also elements of set A--but the only way we have to test the
truth of this assertion is to line up the elements of A with those of B and
see if the correspondance can be made one-to-one.  Shouldn't this have
prevented Cantor from using the one-to-one correspondance of the elements of
set A with the elements of "its proper subset" B as a definition (proof?) of
the infinitude of set A?

If I am being mathematically naive and historically antique, I hope Prof.
Hamilton and other colleagues here will enlighten me now before it is too
late; but if A is, say, the set of integers and B is the set of even
numbers,
how is it that A and B are not what Cantor would call "proper" subsets of C
(the real numbers) unless it is because their elements can NOT be put into
one-to-one correspondance with those of C?

-Bill Everdell, Brooklyn