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

TL;DR

This paper explores Steel's multiverse axioms to identify a preferred universe of set theory, examining the core hypothesis and the potential role of Ultimate-L as the ultimate extension of ZFC.

Contribution

It provides an analysis of the evidential framework for Steel's multiverse, investigates the core of the multiverse, and discusses the justification of Ultimate-L as the preferred universe.

Findings

Large cardinals support the multiverse framework.

The core of the multiverse may be Ultimate-L.

Strategies for justifying V=Ultimate-L are assessed.

Abstract

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.