We will begin with an overview of the hierarchy of definability: from
elementary number theory, through second order arithmetic (analysis),
and on to unrestricted set theory. We will give some interesting
examples which exactly occupy distinguished positions within the
hierarchy, such as arithmetically definable sets which are not
recursive and analytically definable sets which are not Borel. We
will end with a discussion of whether this presentation of the
hierarchy of definability is inevitable, with the conclusion that it
is so.