  The axiom of choice ensures precisely that, in ZFC, every set is projective:
that is, a projective object in the category of sets. In constructive ZF (CZF)
the existence of enough projective sets has been discussed as an additional
axiom taken from the interpretation of CZF in Martin-Loef's intuitionistic type
theory. On the other hand, every non-empty set is injective in classical ZF,
which argument fails to work in CZF. The aim of this paper is to shed some
light on the problem whether there are (enough) injective sets in CZF.
  We show that no two element set is injective unless the law of excluded
middle is admitted for negated formulas, and that the axiom of power set is
required for proving that there are strongly enough injective sets. The latter
notion is abstracted from the singleton embedding into the power set, which
ensures enough injectives both in every topos and in IZF. We further show that
it is consistent with CZF to assume that the only injective sets are the
singletons. In particular, assuming the consistency of CZF one cannot prove in
CZF that there are enough injective sets. As a complement we revisit the
duality between injective and projective sets from the point of view of
intuitionistic type theory.
