FOM: Cantor's subsets
Dean.Buckner at btopenworld.com
Fri Jun 21 13:16:10 EDT 2002
Tho' banished to the outer darkness, I can't help commenting on the latest
postings on this subject (Hamilton). We are given
1) Every A is a B, but not every B is an A
Like Hamilton, let's note in passing the beautiful economy of this
definition, which must have escaped the notice of logicians in the 2,000
year period between Aristotle and Cantor. Also note, like Hamilton, that it
does not contain the word "equinumerous", although it does say that the B's
which are A are not the only B's - there must be others, perhaps you could
say there are "more" than just these B's. Then ask if (1) is consistent
2) For every B, there is an A.
De Morgan, looking at a similar example in his treatise on the Syllogism
(1860) thought not. For each of the B's that are A, there is a matching A
(the B itself). But for the B's that are not A, there is no such match.
Therefore "for those who can see it" (1) and (2) are inconsistent. What
*he* did not see, however, was that his argument involved a hidden
assumption about how many A there are. Let's suppose that
3) No collection of A's contains every A.
Then of course, for every set of B that we consider, though there may not be
"enough" of them that are A to meet (2), it follows from (3) that these
cannot be all the A's. There will be more - "more" than enough in fact to
match the B's that we began with, although they come from outside this
particular set of B's.
That's all I really want to say, except I'm suspicious of the following
(a) that a proper subset can be equinumerous with its parent. The argument
above shows no such thing. It shows that, given (1) and (2), there can be
no set of A that represents "all" the A. There must always be "more".
Indeed, if some proper subset were equinumerous with its parent, there might
not be available any "more" of the A that we need to match the B's, so it
shows the exact reverse!
(b) that certain of our intuitions fail "for infinite sets". Not at all.
It shows De Morgan missed a trick, maybe, but doesn't require a huge leap of
imagination to see how (1), (2) and (3) can all be true. Indeed, it's built
into our idea of the infinite, as of certain kinds of thing are such that
there is no collection representing all such things.
(c) Ditto for Heck's argument about "folk" concepts and their supposed
fallibility. The argument that Heck provided in his posting goes no further
than mine. However, because it is couched in the language of mathematical
set theory, rather than anything Aristotle could have recognised, it seems
to have a mystical significance that, in reality, is entirely absent.
(d) that we can speak of "the" A's, as in the definite noun phrase "the
natural numbers". If this is meant to signify "all" natural numbers, it
fails. Since, for any collection of numbers (B's), there can be found other
numbers (A's). None of our arbitrarily chosen numbers are the A's (the
other numbers), yet, of course, the other numbers are numbers - all numbers
are numbers. There is nothing that satisfies the expression "all the
natural numbers". This is of course is a merely philosophical
consideration, concerning the failure of reference of a purportedly
referring expression of the form "all the F's". Mathematicians can
therefore ignore it as concern about useof English words, semantics or
whatever, and go back to their symbols!
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