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Title: Reflection and potentialism
Author: Roberts, Sam
ISNI:       0000 0004 5918 5572
Awarding Body: Birkbeck, University of London
Current Institution: Birkbeck (University of London)
Date of Award: 2016
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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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available