FOM: RE: Cantor's subsets

[Everdell@aol.com]

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).

<<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.>>

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