FOM: Cantor's Subsets (was RE: precursors of Cantor (&Transfinite Logic))
Dennis E. Hamilton
dennis.hamilton at acm.org
Tue Jun 18 22:58:52 EDT 2002
Well, I did a little homework and found a definition of (proper) subset in
Cantor.
First, an afterthought to my previous posting.
1. I said that there are perfectly straightforward definitions of (proper)
subset in standard treatments of set theory.
Something like,
A is-a-subset-of B
being defined as equivalent to
for all x in A, x is in B
With
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).
2. I forgot that a nice alternative is to say
A is-a-proper-subset-of B
is equivalent to
(A is-a-subset-of B) and ¬(B is-a-subset-of A)
which has a notational purity that appeals to me. It is equivalent to the
formulation in (1), of course.
3. Now for Cantor. The source that I have handy is
[Cantor1915]
Cantor, Georg. Contributions to the Founding of the Theory of Transfinite
Numbers. Translation, Introduction and Notes by Philip E. B. Jourdain.
Open Court (London: 1915). Unabridged republished edition by Dover
Publications (New York: 1955). ISBN 0-486-60045-9 pbk.
I have additional notes on this source at
http://www.infonuovo.com/orcmid/readings/logic.htm#[Cantor1915]
The crux of it is that, according to the Jourdain translation, Cantor used
the notion of a part (Bestandteil) to mean any *other* aggregate whose
elements are also elements of the original one [dh: *emphasis* mine]. This
is on p.86. Cantor did not have our symbology, of course, since Frege,
Peano, and Russell were yet to do their work. So his definition was in
words, apparently very carefully chosen words. An appropriate formulation
is that
A is-part-of B
is equivalent to
(A is-a-subset-of B) and (A ¬= B)
and this is also equivalent to (A is-a-proper-subset-of B) as a consequence
of what it means for one of two *different* sets to contain the other.
Again, there is nothing about numerosity inherent in this notion. That is,
to say "one set has a member that the other does not" is not to say "one set
has more members than the other". I gather this might not be satisfying,
but it is clear to me from what I have read so far that Cantor knew exactly
how to navigate this particular thicket. I would like to separate out any
follow-up on that topic.
It would appear that Cantor's formulation was put forth at least as early as
1878. In the 1895 paper translated by Jourdain, Cantor simply presents the
basic ideas of subsets and equivalence (with respect to cardinality) as part
of the setup for the work he wants to present. He is after "bigger" game,
to be sure.
4. To me, one of the interesting aspects of the standard definition of
subset (not proper subset) is that
A = B
is equivalent to
(A is-a-subset-of B) and (B is-a-subset-of A)
and there is a certain harmony in that too.
-- Dennis
-----Original Message-----
From: Dennis E. Hamilton [mailto:dennis.hamilton at acm.org]
Sent: Monday, June 17, 2002 16:51
To: fom at math.psu.edu
Cc: Everdell at aol.com
Subject: RE: FOM: precursors of Cantor (&Transfinite Logic)
[ ... ]
-----Original Message-----
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
the
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
writers
and philosophers. For the education we get, much thanks.
-Bill Everdell
History
St. Ann's School
Brooklyn, NY
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