STEEL'S PROGRAMME: EVIDENTIAL FRAMEWORK, THE CORE AND ULTIMATE-L

We address Steel's Programme to identify a 'preferred' universe of set theory and the best axioms extending by using his multiverse axioms and the 'core hypothesis'. In the first part, we examine the evidential framework for, in particular the use of large cardinals and of 'worlds' obtained through forcing to 'represent' alternative extensions of. In the second part, we address the existence and the possible features of the core of (where T is +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. To this end, we take into account several strategies, and assess their prospects in the light of 's evidential framework.