FOM: 1-1 correspondence

Dean Buckner Dean.Buckner at
Thu Aug 15 14:12:52 EDT 2002

I have a bucket which I fill with pebbles.  I construct a function as

    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

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