  In this paper we study the notion of strong non-reflection, and its
contrapositive weak reflection. We say theta strongly non-reflects at lambda
iff there is a function F: theta ---> lambda such that for all alpha < theta
with cf(alpha)= lambda there is C club in alpha such that F restriction C is
strictly increasing. We prove that it is consistent to have a cardinal theta
such that strong non-reflection and weak reflection each hold on an unbounded
set of cardinals less than theta .
