Proper Classes: Understanding Ontology Through Paradox. Author: Neil Alexander Barton Institution: University College London Degree: MPhilStud Philosophical Studies 1 I, Neil Alexander Barton, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. 2 Contents 1 Introduction 7 1.1 Classes and Paradox. 7 1.2 The Paradoxes. 8 1.2.1 Russell's Paradox. 8 1.2.2 Cantor's Paradox. 9 1.2.3 The Burali-Forti Paradox. 10 1.2.4 Attacking the Paradoxes. 11 1.2.4.1 What Constitutes a Satisfactory Solution? . 12 1.2.4.2 Locating the Error. 13 1.3 Assumptions of the Thesis. 14 1.3.1 The Iterative Conception of Set. 14 1.3.2 Monist Realism. 22 1.4 A Puzzle for the Two Assumptions. 23 2 Heavyweight Proper Classes 25 2.1 The Heavyweight View. 26 2.1.1 What is the Heavyweight View? . 26 2.1.2 Positive Remarks about the Heavyweight View. 29 2.1.3 A Problem for The Heavyweight View; Why are Proper Classes not Sets? . 30 2.2 Explaining why Proper Classes are not Sets. 32 2.2.1 What a Satisfactory Explanation of why Proper Classes are not Sets Should Achieve. 32 2.2.2 The Non-member Hypothesis. 34 2.2.2.1 Positive Remarks about [NMH]. 35 2.2.2.2 Why [NMH] is Unsatisfactory. 35 2.2.3 The Limitation of Size Hypothesis. 36 2.2.3.1 Positive Remarks about [LSH]. 37 2.2.3.2 Why [LSH] is Unsatisfactory. 38 2.2.4 The Occurrence in the Hierarchy Hypothesis. 41 2.2.4.1 Positive Remarks about [OHH]. 42 2.2.4.2 Why the Heavyweight Theorist Cannot Appeal to [OHH]. 43 2.2.4.3 Conclusion about [OHH]. 50 2.3 Conclusions. 51 3 Modal Views of Proper Classes. 53 3.1 What Parsons Said. 54 3.2 Proper Classes as Modal Functions. 57 3.2.1 The Modal Function View Explained. 57 3.2.2 Evaluation of the Modal Function View. 58 3 3.2.2.1 Initial Positive Remarks. 58 3.2.2.2 Why the Modal Function View Fails. 58 3.3 The Projection View. 60 3.3.1 The Projection View Explained. 60 3.3.2 Evaluation of the Projection View. 61 3.3.2.1 Initial Positive Remarks. 62 3.3.2.2 Why the Projection View is Unsatisfactory. 63 3.4 Conclusions. 65 4 The Plural Account of Proper Classes. 67 4.1 What is the Plural Account? . 68 4.1.1 Plural Reference. 68 4.1.2 Use within Set Theory. 71 4.2 Evaluation of the Plural Account. 73 4.2.1 Positive Remarks about the Plural Account. 73 4.2.2 The Empty Set. 74 4.2.3 Collapse. 75 4.2.4 Superplurals. 79 4.2.4.1 Preserving our Notion of Collection. 83 4.2.4.2 Puzzles for the α-plural Theorist. 83 4.3 Conclusions. 86 5 Conclusion 87 4 List of Figures 1.1 An initial segment of the Cumulative Hierarchy. 16 1.2 Indexing the stages. 16 1.3 The Cumulative Hierarchy . 17 2.1 The structure V. .............................. 52 5 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 find 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 math- ematical 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. 6 Chapter 1 Introduction 1.1 Classes and Paradox. Firstly, what do I mean by class1? We shall see that the notion is more complex than one might immediately expect. Here, however, is a pre-theoretically plausible definition: Definition. A is a class iff A is a collection of 0 or more2 objects such that for any object x it is definite whether or not x is a member of A34. The notion of ‘definite’ given above is one that will recur throughout this thesis. It will therefore serve to be precise about what I mean by the term. Definition. A proposition P is definite iff there is a fact of the matter whether or not P . So, if for any object x it is definite whether or not x is a member of a class Y , then there is a fact of the matter whether or not x is a member of Y . 1I use the term class rather than set as I shall reserve `set' for those classes that form sets under the Iterative Conception of set. 2I use the locution `0 or more' to respect the fact that I consider the empty class to be a class. 3This definition is largely inspired by Cantor; see [Cantor, 1883]. 4A note on terminology; I use the upper-case letters from the start of the English alphabet (`A', `B', `C' etc.) to represent classes, and the lower-case letters from the end of the English alphabet (`x', `y', `z' etc.) to denote objects (including sets). The difference between sets and classes will be explained later. The use of the upper-case letters `X', `Y ', `Z' etc. is reserved for second-order logic, with the exception of `P ' which denotes a proposition. 7 Given the metamathematical use to which we can put reasoning about classes5, we should have cause for concern if our class-theoretical thinking turned out to be inconsistent. As is well known, `na¨ıve' formulations of class theory have just this fatal flaw. 1.2 The Paradoxes. I will examine the following paradoxes: • 1.2.1 Russell's Paradox. • 1.2.2 Cantor's Paradox. • 1.2.3 The Burali-Forti Paradox. 1.2.1 Russell's Paradox. Russell's Paradox has received a great deal of attention6. This is partly due to its historical significance, the paradox was what infamously brought down Frege's original system. However, it is also because the derivation of the contradiction is quite simple and elegant. It will serve first to introduce some notation: Notation. By `fx : φ(x)g' I mean the `class' of all x such that φ(x). The paradox stems from considering the predicate `x is not a member of itself' (in modern notation `x2 = x'). If such a predicate succeeds in defining a class of all objects that are not members of themselves (call it `R') we would have a class such that (8x)(x 2 R $ x2 = x). Assuming that R is an object, we may substitute `R' for `x' in `(8x)(x 2 R $ x2 = x)'. Then we get; 5For excellent reviews of a fraction of its uses see [Giaquinto, 2002] and [Fraenkel et al., 1973]. 6Including from Russell himself. The following quotation speaks volumes about the difficulties he experienced while trying to solve his own paradox (among others); \I was trying hard to solve the contradictions mentioned above. Every morning I would sit down before a blank sheet of paper. Throughout the day, with a brief interval for lunch, I would stare at the blank sheet. Often when evening came it was still empty.....It was clear to me that I could not get on without solving the contradictions, and I was determined that no difficulty should turn me aside from the completion of Principia Mathematica, but it seemed quite likely that the whole of the rest of my life might be consumed in looking at that blank sheet of paper." ([Russell, 1967], p151). 8 R 2 R $ R2 = R which is clearly a contradiction. It is possible to restate the paradox informally. Assuming that R exists, we may ask whether or not it is a member of itself, i.e. is it the case that R 2 R? If it is a member of itself, then it is in the class of all non-self-membered objects (i.e. R), and hence it is not a member of itself. If, on the other hand, it is not a member of itself, it will be a member of the class of all non-self-membered objects i.e. itself, and thus R will be self-membered. Hence R is a member of itself if and only if it is not a member of itself. 1.2.2 Cantor's Paradox. Cantor's paradox is, like Russell's paradox, a very clear and straightforward piece of reasoning.