FOM: precursors of Cantor (&Transfinite Logic)
Dennis E. Hamilton
dennis.hamilton at acm.org
Mon Jun 17 19:51:04 EDT 2002
Without going to historical sources, it seems to me that there are
commonly-used and perfectly clean definitions in standard treatments of set
A is-a-subset-of B
being defined as equivalent to
for all x in A, x is in B
A is-a-proper-subset-of B
being defined as equivalent to
(A is-a-subset-of B) and (there exists an x in B for which x is not in A).
There is no statement about cardinality nor about numerosity.
It seems to me that the difficulties in this discussion aren't about the
set-theoretic definitions, but about problems we have when making particular
interpretations of set theory against our naive (that is, everyday or
natural or common-sense or whatever) ideas about numbers in sets and the
relentless desire to have things make (naive) sense. I had a terrible
struggle with collegiate physics until I gave that up, and I think it
applies here too.
It may be useful to give up trying to force an interpretation we want onto
the theory and let the theory be. Then see how *useful* valid
interpretations of the theory are or are not when applied to other
interesting concepts (like that of countable but not finite sets of
I am reluctant to throw this into the stew, except it does appear to be a
foundational topic concerning (mathematical) theories as standing on their
own. Theories might have useful interpretation in the world but that is
quite different than requiring theories to describe the world as we see it,
and no more. (As far as I have been able to tell, the concern about
material implication in symbolic logic seems to be another one of these.
So, I would say, is Zeno's paradox.)
- - - - - - -
That's my main point. Maybe there is a way to illustrate it without
creating a diversion. If anything from here on gets in the way or is
objectionable, just discard it.
Coming back to the original concern about equinumerosity and proper subsets,
I found it useful to consider the following:
1. Consider the subsets consisting of the even non-negative numbers and the
subset consisting of the odd non-negative integers. These are clearly each
proper subsets of the non-negative integers. They partition the
non-negative integers. Each of them has the same cardinality of the other.
By that I mean that they are each countable and unbounded sets. That the
sets of evens and odds have the same cardinality has nothing to do with one
being a subset of the other.
Notice how little there is to be gained by wondering if there is one more
even number than odd numbers because we started with the "first"
non-negative number in the even set. (Well, there might be some insight to
be gained, but not by assuming there is something wrong.)
2. Keep going. Separate the set of non-negative odd integers into the ones
that are prime (including 1), and those that are composite. Again, we have
a partitioning. They and their "parent" sets have the same cardinality. If
the assumption about the primes is too much of a stretch, separate out all
the ones (odds or evens) that are perfect squares and those that aren't.
It helps to notice how this is useful as long as we don't have it mean too
3. Consider a parent set to be the set of all finite strings over some fixed
alphabet of more than one letter (don't use digits either, not that it
really matters). If including the empty string is uncomfortable, leave it
out. This set has the same cardinality as the set of the natural numbers.
It is trivially enumerable. Then think of the various proper subsets of
this set that are countable and unbounded. They all have the same
cardinality. This sense of equinumerosity is at the foundation of
computation theory and more. We would not abandon it lightly.
The concept of countability (and its companion, enumerability) simply works
too well, and is straightforward enough, so long as we don't struggle to
have it mean too much. Or insist on preserving an interpretation that only
works with subsets of those sets having finite cardinality (or of finite
order as we are prone to say). I have been careful to use "cardinality" in
a way that does not imply "of specific size," though that interpretation is
mainly harmless for finite sets.
We are comfortable with the finite. (For sufficiently small cardinals. ;)
In extending our grasp to the beyond-finite, we cannot expect to bring with
us everything that works with the finite. Requiring that denies us the
treasure we seek. The beyond-finite is elusive and will not surrender to
the clarity and simplicity of the finite. There are things to let go of,
including a degree of certainty that it is something definite we can place
in our grasp. We should not be surprised.
From: owner-fom at math.psu.edu [mailto:owner-fom at math.psu.edu]On Behalf Of
Everdell at aol.com
Sent: Friday, June 14, 2002 20:26
To: fom at math.psu.edu
Subject: Re: FOM: precursors of Cantor (&Transfinite Logic)
On June 10, 2002, Dean Buckner wrote:
[ ... ]
What is a "subset"? I once searched my Cantor for a definition and found
none. Usually when the mathematical writer feels a certain slipperiness in
the concept, he or she adds the word "proper," as Cantor did (in German),
although "proper" is not defined--and neither have I seen any referent for
"improper subset." It seems to me, therefore, that the proof that a set is
"equinumerous" (that's not the same as "equipollent," right?) with one of
set's subsets, especially an infinite subset, assumes one of the premises
necessary to the proof, viz., that the "subset" in question is somehow
"included" in the set, despite they're being the same size.
[ ... ]
So I shall not join the apostates like Bertrand Russell in outer darkness,
and I expect other historians like me will continue to lurk here with
and philosophers. For the education we get, much thanks.
St. Ann's School
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