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

Steel's Programme: Evidential Framework, the Core and Ultimate-L