  We introduce the notion of a critical cardinal as the critical point of
sufficiently strong elementary embedding between transitive sets. Assuming the
axiom of choice this is equivalent to measurability, but it is well-known that
choice is necessary for the equivalence. Oddly enough, this central notion was
never investigated on its own before. We prove a technical criterion for
lifting elementary embeddings to symmetric extensions, and we use this to show
that it is consistent relative to a supercompact cardinal that there is a
critical cardinal whose successor is singular.
