  These notes aim to give a gentle account to one approach to the theory of
finite sets without making use of the natural numbers. They were written to be
used as the basis for a student seminar. There are no real prerequisites except
for a certain familiarity with the kind of mathematics seen in the first couple
of years of a university mathematics course. The definition of being finite
employed in these notes is usually called Kuratowski-finiteness and it is
essentially that employed by Whitehead and Russell in Principia Mathematica.
  This is a revised and extended version of a paper with the same title from
November 2015
