  We address Steel's Programme to identify a 'preferred' universe of set theory
and the best axioms extending ZFC by using his multiverse axioms MV and the
'core hypothesis'. In the first part, we examine the evidential framework for
MV, in particular the use of large cardinals and of 'worlds' obtained through
forcing to 'represent' alternative extensions of ZFC. In the second part, we
address the existence and the possible features of the core of MV_T (where T is
ZFC+Large Cardinals). In the last part, we discuss the hypothesis that the core
is Ultimate-L, and examine whether and how, based on this fact, the Core
Universist can justify V=Ultimate-L as the best (and ultimate) extension of
ZFC. To this end, we take into account several strategies, and assess their
prospects in the light of MV's evidential framework.
