  It is a striking fact from reverse mathematics that almost all theorems of
countable and countably representable mathematics are equivalent to just five
subsystems of second order arithmetic. The standard view is that the
significance of these equivalences lies in the set existence principles that
are necessary and sufficient to prove those theorems. In this article I analyse
the role of set existence principles in reverse mathematics, and argue that
they are best understood as closure conditions on the powerset of the natural
numbers.
