FOM: A Puzzle

[Dean Buckner]

We can place any suitable collection of names in a one-one correspondence.
For example

Clemens, Poe, Twain
Austen, Bronte, Carlyle

What is the guarantee that the things they name have the same number?  In
this case there is no such, the first series naming two authors, the other,
three.

We need as a necessary condition, that neither set of names contain a
(proper) subclass of names designating a collection with the same number as
the collection designated by the set itself.  But consider

(N) "1, 2, 3, 4 ..."
(E)  "2, 4, 6, 8 ..."

The (N) series of names  "1", "2", "3" has been placed in correspondence
with the (E) series.  Suppose each collection of names is infinite, and that
the collection signified by (N) is equinumerous with what is signified by
(E).  Then (N) will not contain a subset designating a similarly-numbered
collection.  But of course it will, since (N) must contain (E).  Either the
collection cannot be infinite, or what (N) designates has a different number
from what (E) designates.  Yet each numeral in each series designates a
unique object!

This depends on a number of assumptions, but I will let the brains at FOM
tease these out.  Note, before hasty answers, I am not (necessarily) denying
that the *names* in (N) and (E) are equinumerous.  It is what they name, I
am worried about.



Dean Buckner
4 Spencer Walk
London, SW15 1PL
ENGLAND

Work 020 7676 1750
Home 020 8788 4273