# [FOM] Finite sets: two references

Dean Buckner d3uckner at btinternet.com
Fri Feb 24 13:55:15 EST 2006

```Allen:
>  (as long as we .  are willing to assume the existence of at least
two urelements  to get things started) a theory of plurimembered sets (=
sets each of which has two or more members) can be constructed which has
the same
MATHEMATICAL content as the usual set theory.
>

Why do we need to do this?  On the assumption that 'there are F's' leads
to a contradiction, which of the two following proofs is simpler?

1.  There are F's
2.  (From 1) derive a contradiction
3.  There are no F's

-------------

1.  The set {x: x is F}is non-empty
2.  (From 1) there are F's
3.  (From 2) derive a contradiction
4.  The set {x: x is F}is empty
5.  There are no F's

Or do we need it for saying that the intersection {x: x is F} and {x: x
is G} is empty?  But why?  Why not just say that there are no F's that
are G, or that the description 'the F's which are G' refers to nothing?
What is the empty set for?

```