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Modal structuralism and reflection

Roberts, Sam Royston
Journal article; PublishedVersion; Peer reviewed
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modal_structuralism_and_reflection.pdf (368.9Kb)
Year
2018
Permanent link
http://urn.nb.no/URN:NBN:no-70796

CRIStin
1594064

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Appears in the following Collection
  • Institutt for filosofi, ide- og kunsthistorie og klassiske språk [307]
  • CRIStin høstingsarkiv [15141]
Original version
The Review of Symbolic Logic. 2018, DOI: https://doi.org/10.1017/S1755020318000138
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.
 
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