A Reconstruction of Steel’s Multiverse Project

Bulletin of Symbolic Logic 26 (2):118-169 (2020)
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Abstract

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 isolate 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

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Author Profiles

Penelope J. Maddy
University of California, Irvine
Toby Meadows
University of California, Irvine

References found in this work

Platonism and Anti-Platonism in Mathematics.Mark Balaguer - 1998 - Bulletin of Symbolic Logic 8 (4):516-518.
The set-theoretic multiverse.Joel David Hamkins - 2012 - Review of Symbolic Logic 5 (3):416-449.
Set-theoretic geology.Gunter Fuchs, Joel David Hamkins & Jonas Reitz - 2015 - Annals of Pure and Applied Logic 166 (4):464-501.

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