  In two recent papers (math.LO/0003164 and math.LO/0003165) we answered a
question raised in the book by Eklof and Mekler (p. 455, Problem 12) under the
set theoretical hypothesis of diamondsuit_{aleph_1} which holds in many models
of set theory, respectively of the special continuum hypothesis (CH). The
objects are reflexive modules over countable principal ideal domains R, which
are not fields. Following H.Bass, an R-module G is reflexive if the evaluation
map s:G ---> G^{**} is an isomorphism. Here G^*=Hom(G,R) denotes the dual
module of G. We proved the existence of reflexive R-modules G of infinite rank
with G not cong G+R, which provide (even essentially indecomposable) counter
examples to the question mentioned above. Is CH a necessary condition to find
`nasty' reflexive modules? In the last part of this paper we will show
(assuming the existence of supercompact cardinals) that large reflexive modules
always have large summands. So at least being essentially indecomposable needs
an additional set theoretic assumption. However the assumption need not be CH
as shown in the first part of this paper. We will use Martin's axiom to find
reflexive modules with the above decomposition which are submodules of the
Baer-Specker module R^omega.
