FOM: RE: Transfinite Logic
Dean Buckner
Dean.Buckner at btopenworld.com
Sun Jun 9 06:59:50 EDT 2002
On the theorem that the set of all natural numbers {0,1,...} is equinumerous
with the set of all even numbers (0,2,...}, Richard Heck writes "There is no
questioning this result. It is a totally straightforward, and extremely
simple, theorem"
Agreed, if it is meant that (in plain English) every number has a double,
and every double is the double of one, and no more than one number, which
follows entirely from the definitions Heck has set out.
I question his definitions. If every number has a double, and every double
is the double of just one number, does it follow that there are "as many"
doubles (even numbers) as singles (integers), given the plain English
meaning of "as many"? . Is it correct to define "as many" or
"equinumerous" using the idea of 1-1 correspondence? My argument is about
use of English, not mathematics!
The argument usually then turns to the challenge of defining "as many as".
This brings us right to the very beginning, when I argued (in some earlier
postings) that can think of number as satisfying n-place predicates such as
"x is a different thing from y", which is satisfied by Abbott and Costello,
Oliver and Hardy, but not Clemens and Twain, and not the Three Stooges.
Any two collections are equinumerous (have as many objects as each other)
when there is such a predicate they both satisfy. Defined in this way, no
proper subset can be equinumerous with its parent. The parent, by
definition, contains objects "different" from any of those in the subset,
hence the parent cannot satisfy the same n-place predicate as the subset.
It's not open now to argue that "it's different for infinite sets" as many
of my offline correspondents have tried to do. Having defined "proper
subset" and "equinumerous" in a way that no proper subset can be
equinumerous with its parent, it's not open to say this any more. The
correct approach (which Heck alone has sensibly adopted) is to challenge
this definition
of "as many as".
Except I question that he has successfully done so. My theory, if you like,
is that natural language has built into it a system of numbering, and a
concept of number that is at odds with "transfinite" ideas about number.
This is why - given that natural language underlies the way we all think (or
at least learn to think as children) - people sometimes find these ideas
"difficult" (as Heck points out).
This may also be why it is difficult to explain transfinite ideas, as
embedded in mathematical symbolism, since to "explain" anything is to
translate such symbols into plain everyday language. If (as I've argued) we
can't do this, then there's a problem. Particularly, as we saw, for writers
of elementary textbooks.
Concerning the misunderstandings between philosophers and mathematicians on
this subject, there is a good paper (by a philosopher) on the following
page:
http://community.middlebury.edu/~arthur/LeibCant.pdf
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