  In another paper (Butterfield 2011), one of us argued that emergence and
reduction are compatible, and presented four examples illustrating both. The
main purpose of this paper is to develop this position for the example of phase
transitions. We take it that emergence involves behaviour that is novel
compared with what is expected: often, what is expected from a theory of the
system's microscopic constituents. We take reduction as deduction, aided by
appropriate definitions. Then the main idea of our reconciliation of emergence
and reduction is that one makes the deduction after taking a limit of an
appropriate parameter $N$. Thus our first main claim will be that in some
situations, one can deduce a novel behaviour, by taking a limit $N\to\infty$.
Our main illustration of this will be Lee-Yang theory. But on the other hand,
this does not show that the $N=\infty$ limit is physically real. For our second
main claim will be that in such situations, there is a logically weaker, yet
still vivid, novel behaviour that occurs before the limit, i.e. for finite $N$.
And it is this weaker behaviour which is physically real. Our main illustration
of this will be the renormalization group description of cross-over phenomena.
