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MODAL STRUCTURALISM AND REFLECTION

Published online by Cambridge University Press:  14 June 2018

SAM ROBERTS
SAM ROBERTS*
Affiliation:
Department of Philosophy, IFIKK, University of Oslo
*
*DEPARTMENT OF PHILOSOPHY IFIKK UNIVERSITY OF OSLO POSTBOKS 1020 BLINDERN 0315 OSLO NORWAY E-mail: [email protected]URL: http://samrroberts.net
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Abstract

Modal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the article, I look at the prospects for supplementing MSST with a modal structural reflection principle, as suggested in Hellman (2015). I show that Hellman’s principle is inconsistent (Theorem 5.32), and argue that modal structural reflection principles in general are either incompatible with modal structuralism or extremely weak.

Keywords

03A0503E5503E3097E2097E3000A3003B45structuralismmodal structuralismset theorymodal logicplural logiclarge cardinalsreflection principles
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 4 , December 2019 , pp. 823 - 860
Copyright
Copyright © Association for Symbolic Logic 2018 

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