Reflection and potentialism

Roberts, Sam (2016) Reflection and potentialism. PhD thesis, Birkbeck, University of London.

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Abstract

It was widely thought that the paradoxes of Russell, Cantor, and Burali-Forti had been solved by the iterative conception of set. According to this conception, the sets occur in a well-ordered transfinite series of stages. On standard articulations – for example, those in Boolos (1971, 1989) – the sets are implicitly taken to constitute a plurality. Although sets may fail to exist at certain stages, they all exist simpliciter. But if they do constitute a plurality, what could stop them from forming a set? Without a satisfactory answer to this question, the paradoxes threaten to reemerge. In response, it has been argued that we should think of the sets as an inherently potential totality: whatever things there are, there could have been a set of them. In other words, any plurality could have formed a set. Call this potentialism. Actualism, in contrast, is the view that there could not have been more sets than there are: whatever sets there could have been, there are. This thesis explores a particular consideration in favour of actualism; namely, that certain desirable second-order resources are available to the acutalist but not the potentialist. In the first part of chapter 1 I introduce the debate between potentialism and actualism and argue that some prominent considerations in favour of potentialism are inconclusive. In the second part I argue that potentialism is incompatible with the potentialist version of the second-order comprehension schema and point out that this schema appears to be required by strong set-theoretic reflection principles. In chapters 2 and 3 I explore the possibilities for reflection principles which are compatible with potentialism. In particular, in chapter 2 I consider a recent suggestion by Geoffrey Hellman for a modal structural reflection principle, and in chapter 3 I consider some influential proposals by William Reinhardt for modal reflection principles.

Item Type: Thesis (PhD)
Copyright Holders: The copyright of this thesis rests with the author, who asserts his/her right to be known as such according to the Copyright Designs and Patents Act 1988. No dealing with the thesis contrary to the copyright or moral rights of the author is permitted.
School/Department: School of Social Sciences, History & Philosophy > Philosophy
Depositing User: ORBIT Editor
Date Deposited: 03 Mar 2016 14:09
Last Modified: 03 Mar 2016 14:17
URI: http://bbktheses.da.ulcc.ac.uk/id/eprint/171

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