FOM: RE: Cantor's subsets

Everdell@aol.com Everdell at aol.com
Sun Jun 23 16:51:13 EDT 2002

```On Monday, June 17, 2002, Dennis Hamilton replied to my question: What is a
subset?:

<<Without going to historical sources, it seems to me that there are
commonly-used and perfectly clean definitions 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).

<<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.>>

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).
Hamilton writes:

<<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.>>

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

```