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Reflections on Skolem's Paradox

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Reflections on Skolem's Paradox Reflections on Skolem’s Paradox Timothy Bays Logical investigations can obviously be a useful tool in philosophy. They must, however, be informed by a sensitivity to the philosophical significance of the formalism and by a generous admixture of common sense, as well as a thorough understanding both of the basic concepts and of the technical details of the formal material used. It should not be supposed that the formalism can grind out philosophical results in a manner beyond the capacity of ordinary philosophical reasoning. There is no mathematical substitute for philosophy. Saul Kripke: Is there a Problem about Substitutional Quantification? i Contents Introduction 1 1 A Simple Paradox 4 1.1 A Simple Paradox I . 4 1.2 A Simple Paradox II . 11 1.3 Solving Skolem’s Paradox . 20 2 Some Complicated Paradoxes 30 2.1 Transitivity . 31 2.2 Elementarity . 37 2.3 Membership . 40 2.4 Conclusion . 47 3 Two Philosophical Objections 49 3.1 Naive Prattle . 49 3.2 Axioms and Mathematical Content . 62 3.3 Conclusion . 77 4 On Putnam and his Models 79 4.1 Two Preliminary Clarifications . 79 4.2 Putnam’s Argument . 81 4.3 The Mathematics of Premise 1 . 84 4.4 The Philosophy of Premise 2 . 89 4.5 Some Connections . 104 4.6 Conclusion . 106 Bibliography 111 ii Introduction In 1922, Thoraf Skolem published a paper entitled “Some Remarks on Axiomatized Set Theory.” The paper presents a new proof of a model-theoretic result originally due to Leopold L¨owenheim, and it then discusses some of the philosophical implications of this result. In the course of this latter discussion, the paper introduces a model-theoretic puzzle that has come to be known as “Skolem’s Paradox.” The present dissertation focuses on this paradox. Skolem’s Paradox involves a seeming conflict between two theorems of modern logic: Cantor’s theorem from set theory and the L¨owenheim-Skolem theorem from model theory. Cantor’s theorem says that there are uncountable sets—sets that are too big to be put into any one-to-one correspondence with the natural numbers. The L¨owenheim-Skolem theorem says that if a collection of first-order sentences has a model, then it has a model which is only countable. Skolem’s paradox arises when we note that the axioms of set theory can themselves be written as first-order sentences: how can the very axioms which prove the existence of uncountable sets be satisfied by a merely countable model? Ever since Skolem formulated this paradox, philosophers have argued that issues of deep philosophical importance hang on the paradox’s resolution. Skolem himself claimed that the paradox shows that set theory provides an inadequate foundation for mathematics. Later authors have used the paradox to argue that “every set is countable from some perspective” (Wang), that substitutional quantification is equivalent to objectual quantification (Fine), or that Quine’s theory of ontological reduction is hopelessly flawed (Grandy and Chihara). Most recently, Hilary Putnam has claimed that the paradox has, in his words, “profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language.” It should be fairly clear, even at this preliminary stage, that there are two different things which these kinds of arguments need to do. First, they need to show that Skolem’s Paradox really is a paradox! That is, they need to show that Skolem’s Paradox exposes a genuine tension between Cantor’s theorem and the L¨owenheim-Skolem theorem and that eliminating this tension requires a modification in our initial views about, e.g., set theory. Once they have done this, they can go ahead and explain how their more dramatic— or, at least, more explicitly philosophical—conclusions are supposed to follow. This dissertation focuses almost exclusively on the first half of this project—i.e., the half which tries to expose an initial tension between Cantor’s theorem and the L¨owenheim-Skolem theorem. I argue that, even on quite naive understandings of set theory and model theory, there is no such tension. Hence, Skolem’s 1 Paradox is not a genuine paradox, and there is very little reason to worry about (or even to investigate) the more extreme consequences that are supposed to follow from this paradox. The heart of my solution to Skolem’s Paradox can be found in chapter 1. In 1.1 and 1.2, I formulate a relatively simple version of this paradox. In doing so, I attempt to disentangle the different roles which set theory, model theory and philosophy play in the paradox. I also isolate the features of the paradox which make it feel paradoxical in the first place.1 Finally, in 1.3, I explain why this formulation of Skolem’s Paradox does not constitute a genuine “paradox” after all. Very roughly, my explanation goes as follows. In 1.2.1, I isolate six sentences (or, in two cases, formulas) that live in the near neighborhood of the English phrase “x is uncountable.” In 1.2.2, I show that Skolem’s Paradox turns on an equivocation between two of these sentences. Then, in 1.3, I present a series of four arguments which show why this equivocation cannot be justified. At the end of the chapter, therefore, I conclude that this version of Skolem’s Paradox exposes no real tensions between Cantor’s theorem and the L¨owenheim-Skolem theorem. One way to avoid the argument of chapter 1 would be by reformulating Skolem’s Paradox so as to rely on more sophisticated mathematics—i.e., not just on the L¨owenheim-Skolem theorem. In chapter 2, I consider three examples of this strategy: one which involves transitive models, one which involves elementary submodels of Vκ for some inaccessible κ, and one which involves theorems about permutations of the set- theoretic universe.2 In the long run, I argue that all of these formulations fail and that, although it may take some mathematical drudge-work to discover this fact, they fail for the same reason that the original formulation of Skolem’s Paradox fails. At the end of the day, then, I conclude that there is nothing even in the vicinity of Skolem’s Paradox which exposes a genuine tension between Cantor’s theorem and the L¨owenheim-Skolem theorem. In the course of this argument in chapter 2, another interesting feature of Skolem’s Paradox comes to light. Philosophers have often argued that Skolem’s Paradox is due, in essence, to the specific way that first-order models interpret existential quantifiers. More specifically, there is one particular quantifier in the formal expression corresponding to the English phrase “x is uncountable” which tends to get the “blame” for Skolem’s Paradox. In 2.2 and 2.3, I show that this analysis is, at least partially, misguided. There are many different symbols which can “take the blame” for Skolem’s Paradox, and it’s technically important to understand which ones can (and cannot) do this in specific cases. In chapter 3, I examine a second puzzle concerning Skolem’s Paradox. Ever since the 20’s, philosophers have known that there is a relatively simple technical solution to Skolem’s Paradox. Nevertheless, philoso- phers have continued to find the paradox tempting and have, in many cases, claimed that it poses a genuine philosophical problem (i.e., as discussed at the beginning of this introduction). Why, then, have so many philosophers found the technical “solution” to Skolem’s Paradox less than adequate? 1See also the first few pages of 1.3.1 for a discussion of the “feel” of Skolem’s Paradox. 2See the relevant sections of chapter 2 for definitions of, e.g., “transitive,” “elementary,” and “inaccessible.” 2 In chapter 3, I suggest two reasons for this attitude. The first involves a general suspicion of the mathe- matical machinery that is used to formulate the technical solution, while the second argues that the technical solution overlooks something important about the role of axiomatization in mathematics. In 3.1, I formulate a version of the first position and then give a reply to it. Basically, I claim that there may well be something to worry about here, but that the worries in question are so fundamental that they reduce Skolem’s Paradox itself to a triviality. If these worries are well-founded, then set-theoretic problems appear immediately, and Skolem’s Paradox functions as mere technical window-dressing. In 3.2, I tackle the second position. I begin by isolating the exact role which axiomatization would need to play in mathematics if Skolem’s Paradox were to be a genuine problem. I then give two different arguments against the idea that axioms play (or even could play) this particular role. In the end, I conclude that neither of the arguments against the technical solution to Skolem’s Paradox can really be sustained. Since these are the only arguments which threaten my own solution to Skolem’s Paradox—i.e., the solution given in chapters 1 and 2—I conclude that my solution remains persuasive. Finally, in chapter 4, I turn away from generic formulations of Skolem’s Paradox to examine Putnam’s so-called “model-theoretic argument against realism.” I show that Putnam’s argument involves mistakes of both the mathematical and the philosophical variety, and that these two types of mistake are closely related. Along the way, I clear up some of the mutual charges of question-begging which have characterized discussions between Putnam and his critics. In the end, I conclude that Putnam’s use of Skolem’s Paradox involves far less originality than is sometimes supposed and that his “model-theoretic argument against realism” is simply a failure.
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