FOM: 1-1 correspondence
Dean Buckner
Dean.Buckner at btopenworld.com
Thu Aug 15 14:12:52 EDT 2002
I have a bucket which I fill with pebbles. I construct a function as
follows
f(p1) = {p1}
f(p2) = {p1,p2}
f(p2) = {p1,p2,p3}
and so on
Where p1 is the first pebble in the bucket, p2 the second and so on. I map
each pebble I have just put in the bucket, with the set of pebbles that are
already in the bucket (including that one).
Once I stop putting pebbles in, it follows there is one pebble in the bucket
px such that
f(px) = S
where S is the entire set of pebbles. So, can we define a finite set as one
where there exists such a mapping (which is easily defined recursively)?
It can be shown, I think, that no such set can be equinumerous with any
proper subset. If so, it's a neat definition of finitude, because it
captures our intuition about finite sets - if we count out the pebbles we
come to the end at some point - without having to appeal to the existence of
any such process. All that is required is that there be a mapping of the
right kind.
But is this correct? Would be grateful for views.
Dean Buckner
Putney
London
ENGLAND
Work 020 7676 1750
Home 020 8788 4273
More information about the FOM
mailing list