We address Steel's Programme to identify a 'preferred' universe of set theory (and the best theory extending ZFC), in the context of the multiverse axioms MV and of 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 and bind together alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MVT (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 the Core Universist might use salient facts about the core provable in MVT to settle on 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.We may re-state Weak Absolutism in a slightly different, but maybe more perspicuous, way, as follows:Core Universism. Set theory is the theory of multiple (set-theoretic) universes, each of which contains a unique core universe, which has a better claim to be seen as the 'preferred' universe of sets than any other universe.To state very quickly the main differences between the two positions: the Classic Universist may be standardly characterised as someone believing that our intuition of sets, or the 'concept of set' itself, will provide us with a unique, consistent extension of the ZFC axioms which will uniquely fix the truth-value of the undecidable statements. 11 By contrast, the Core Universist may be characterised as someone who takes all alternative 'universes' as equally legitimate; however, such a Universist will also hold that each universe (or, if you wish, theory) contains 'traces' of a single, 'preferred' universe, and much of the value of the position consists in showing that the claim is true, that is, that a core universe is really detectable within the multiverse itself. Of course, the Core Universist also expects to be able to describe the properties of the core in a satisfactory way.In order to attain a reduction of set-theoretic incompleteness, the Classic Universist will suggest further exploration of our intuitions about sets, or sharpening of the concept of set, whereas the Core Universist will suggest further exploration of the properties of the core through the multiverse axioms. Now, it is clear that Steel's Programme advocates the Core Universist's standpoint, and, therefore, has deep implications on the preferability of Core over Classic Universism: as already said, the Programme, if successful, would lend support to Core Universism. Indeed, the Core Universist's construal of the Programme's goals and results could be condensed as follows: 'Non-pluralism about set-theoretic ontology cannot be correct, as we are aware of the existence of many alternative universes (as well as of alternative theories extending ZFC). However, given a suitable version of the multiverse, one resting upon significant bits of current s...