A RECONSTRUCTION OF STEEL'S MULTIVERSE PROJECT

Penelope Maddy, Toby Meadows
2020 Bulletin of Symbolic Logic  
This paper reconstructs Steel's multiverse project in his 'Gödel's program' (Steel [2014]), first by comparing it to those of Hamkins [2012] and Woodin [2011], then by detailed analysis what's presented in Steel's brief text. In particular, we reconstruct his notion of a 'natural' theory, describe his multiverse axioms and his translation function, and assess the resulting status of the Continuum Hypothesis. In the end, we reconceptualize the defect that Steel thinks CH might suffer from and
more » ... late what it would take to remove it while working within his framework. As our goal is to present as coherent and compelling a philosophical and mathematical story as we can, we allow ourselves to augment Steel's story in places (e.g., in the treatment of Amalgamation) and to depart from it in others (e.g., the removal of 'meaning' from the account). The relevant mathematics is laid out in the appendices. The stubborn recalcitrance of some independent set-theoretic statements, most prominently the Continuum Hypothesis (CH), and the proliferation of powerful techniques for generating new models have led some observers to champion a stark revision in our understanding of the set-theoretic project: the goal isn't to 5 develop a theory, as complete as possible, describing a single universe of sets; rather, the target is an array of universes, a multiverse. Several such theories have been proposed, and the general idea is now prevalent enough to have made its way into the prose of at least one textbook (Weaver [2014] ). To take the example of CH, most such theories posit an array of universes with CH true in some 10 and false in others, which is taken to show that it has no determinate truth value, that efforts to settle it definitively, one way or the other, are misguided. Against this backdrop, John Steel's approach is particularly intriguing: he offers his multiverse theory instead as a means toward assessing CH, of exploring whether or 1 This is a "preproof" accepted article for The Bulletin of Symbolic Logic. This version may be subject to change during the production process.
doi:10.1017/bsl.2020.5 fatcat:qeovhbt65ba4bcleuucjjb45cm