We consider the sublanguages of Plotkin's PCF obtained by imposing some bound
k on the levels of types for which fixed point operators are admitted. We show
that these languages form a strict hierarchy, in the sense that a fixed point
operator for a type of level k can never be defined (up to observational
equivalence) using fixed point operators for lower types. This answers a
question posed by Berger. Our proof makes substantial use of the theory of
nested sequential procedures (also called PCF B\"ohm trees) as expounded in the
recent book of Longley and Normann
