Barton, NA; (2013) Proper Classes: Understanding Ontology Through Paradox. Masters thesis , UCL (University College London).

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Abstract

Engaging in the philosophical debate surrounding proper classes is a tough task. The waters are both muddy and treacherous. In particular, there is no full and rigorous analysis of the logical geography of the area; what one should accept if one makes certain assumptions about ontology. The water would be substantially cleaned up (if not made any less treacherous) by a thorough and comprehensive treatment of just this. This thesis can be seen as one part of a comprehensive study. The methodology is to precisely state some ontological assumptions, and then to examine how one should characterise proper classes given these assumptions. In Chapter 1 I outline the paradoxes I will be considering and the assumptions I am making. I suggest that a better philosophical understanding of ontology would allow us to motivate a solution to the paradoxes. In Chapter 2 I consider (and reject) just such an ontological thesis; the conception that proper classes are ontologically `heavyweight'. In Chapter 3 I examine some views that attempt to characterise proper classes using modal resources. I then give reasons as to why I nd these standpoints unsatisfactory. Chapter 4 provides an analysis and defence of the stance that proper classes are artefacts of plural reference. The thesis of this work is that if one holds a certain kind of realism about mathematical ontology, then (under the Iterative Conception of set) proper classes do not exist. However, reference to proper classes can be understood as plural reference to some sets.

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