  We investigate the set S(R) of shift-isomorphism classes of semidualizing
R-complexes, ordered via the reflexivity relation, where R is a commutative
noetherian local ring. Specifically, we study the question of whether S(R$ has
cardinality 2^n for some n. We show that, if there is a chain of length n in
S(R) and if the reflexivity ordering on S(R) is transitive, then S(R) has
cardinality at least 2^n, and we explicitly describe some of its
order-structure. We also show that, given a local ring homomorphism f: R\to S
of finite flat dimension, if R and S admit dualizing complexes and if f is not
Gorenstein, then the cardinality of S(S) is at least twice the cardinality of
S(R).
