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Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility
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Stewart Shapiro

2007
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The Western Ontario Series in Philosophy of Science
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Background. What and why? This paper is a contribution to an ongoing program in the philosophy of mathematics that began with Crispin Wright ([1983]), was bolstered by Bob Hale ([1987]), and now continues through many extensions, objections, and replies to objections. The neo-logicist plan is develop branches of established mathematics using abstraction principles in the form:~a~b (((a)=((b) E(a,b)), where a and b are variables of a given type, typically ranging over either individual objects
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... properties; ( is a higher-order operator, denoting a function from items in the range of the given type to objects in the range of the first-order variables; and E is an equivalence relation over items of the given type. The main thesis of neo-logicism concerns the epistemic status of some abstraction principles. The neo-logicist claims that certain abstraction principles are, or are like, implicit definitions, and true by stipulation. Thus, the program provides an epistemological foundation for the consequences of those principles. The collection Hale and Wright ([2001] ) contains detailed articulations of the goals and purposes of the neo-logicist program, and of the proposed status of certain abstraction principles (see especially Wright [1997] ). We need not broach the exegetical question of the extent to which the arch-logicist Gottlob Frege accepted the orientation of neo-logicism. Frege ([1884], [1893]) did employ at least three abstraction principles. One of them, used for illustration, comes from geometry: The direction of l 1 is identical to the direction of l 2 if and only if l 1 is parallel to l 2 . Frege's second abstraction principle was dubbed N = in Wright ([1983]) and is now called Hume's principle: where FG is an abbreviation of the second-order statement that there is a relation mapping the F's one-toone onto the G's. Hume's principle thus states that the number of F is identical to the number of G if and only if the F's are equinumerous with the G's. Unlike the principle concerning directions, this abstraction is second-order, since the relevant variables, F, G range over concepts or properties of whatever is in the range of the first-order variables. Hume's principle also differs from the direction principle in that the equivalence on the right hand side contains only logical terminology (assuming that second-order variables and quantifiers are logical). Frege ([1884]) contains the essentials of a derivation of the Peano postulates from Hume's principle. This deduction, now called Frege's theorem, reveals that Hume's principle entails that there are infinitely many natural numbers. On the neo-logicist program, then, Hume's principle provides an epistemological foundation for arithmetic. Frege's third abstraction is the infamous Basic Law V: Like Hume's principle, Basic Law V is a second-order, logical abstraction, but unlike Hume's principle, it is inconsistent. An essential part of the ongoing neo-logicist program is to articulate principles that indicate which abstraction principles serve as legitimate epistemic foundations for mathematical theories. A proposed abstraction principle must be consistent, of course, but consistency is not sufficient. There are abstraction principles which are individually consistent, but are mutually incompatible (see Heck [1992] and Weir [2001]). We do not enter into the subtle details of this issue here. An important item on the neo-logicist agenda is to extend the success of Frege's theorem to other, richer branches of mathematics. The idea is to formulate acceptable abstraction principles that recapture -3those branches, in much the same way that Hume's principle recaptures arithmetic. Hale ([2000]) attempts this for real analysis, as does Shapiro and Wright ([2002]), which includes a brief account of complex analysis. The purpose of this article is assess the prospects for a neo-logicist recapture of set theory. I suggest that set theory is a particularly important case, if neo-logicism is to dovetail with the full range of contemporary and historical mathematics. The notion of 'set' plays a central role within many branches, and set theory itself has come to enjoy a foundational significance. Since virtually every extant mathematical structure can be modeled in the set-theoretic hierarchy, set theory provides a natural setting for comparing and relating different mathematical structures. Moreover, the set-theoretic hierarchy is the de facto arena for resolving existence, or consistency issues within mathematics, and the set-theoretic hierarchy provides a natural setting for comparing and relating different mathematical structures. Thus, if the neo-logicist fails to capture a reasonably rich set theory, then the program has left out a crucial part of contemporary mathematics, one with special foundational significance. One the other hand, someone might think that set theory is already implicit in the neo-logicist systems. The extant developments-including Frege's theorem deriving the Peano postulates from Hume's principle-make essential use of second-order logic. Officially, the monadic higher-order variables range over properties or propositional functions of whatever is in the range of the first-order variables, but properties have structural affinities with sets. Bertrand Russell ([1903], p. 13), for example, wrote that '. . . the study of propositional functions appears to be strictly on a par with that of classes, and indeed scarcely distinguishable therefrom'. During his later no-class period, Russell ([1993], Chapter 18) proposed to eliminate talk of classes by replacing variables ranging over classes with higher-order variables. In another context, W. V. O. Quine ([1986], Chapter 5) famously argues that second-order logic is not logic at all, but is set theory in disguise, a 'wolf in sheep's clothing'. So one might think that some set -4theory has already been smuggled into the neo-logicist program, lying disguised in the higher-order logic. Notice, for example, that Hume's principle is equi-consistent not with classical arithmetic, but with classical analysis, the theory of natural numbers and sets of natural numbers (Boolos [1987] ). We need not broach the border dispute concerning whether second-order logic is properly part of logic or is mathematics in disguise. The theme of both logicism and neo-logicism is that mathematics and logic are intertwined. The underlying issues here are epistemic. Our Quinean might argue that Frege's theorem only shows how to derive the Peano postulates from Hume's principle plus some set theory. When put this way, the result has no deep philosophical significance, since we already know that set theory is mathematically richer than arithmetic, and can serve as a foundation for it. Frege's theorem merely recounts what we already know, that we can recapture arithmetic from the basic principles of set theory. This particular charge can be rebutted with a study of the particular axioms and rules of secondorder logic invoked in Frege's theorem. Do those particular principles presuppose a substantial set theory? Typically, the study focuses on the instances of the comprehension scheme, and the use of the abstraction operators, often focusing on how impredicative those are (see, for example, Wright [1998]). We can safely bracket this foundational issue here. Even if second-order logic is a disguised set theory, it is a rather weak one. Let us call the items in the range of the first-order variables "objects". Monadic second-order variables range over properties-or perhaps sets-of those objects. We cannot take all of these properties or sets to be objects in the range of the first-order variables, on pain of contradiction. Cantor's theorem is that there are more properties or sets than objects. General second-order variables range over analogues of sets of n-tuples of objects. But that is the limit of second-order logic. If the neologicist ventures into third-order logic, then she has analogues of sets of sets of objects, and fourth-order logic brings analogues of sets of sets of sets of objects. Perhaps the neo-logicist would be well-advised to keep the "order" low, to minimize the charge of smuggling the mathematics in. However, even if the neo-devoting several sessions to this project. Special thanks to

doi:10.1007/978-1-4020-4265-2_18
fatcat:n4q7zvaltzgihajqmblrrzohsy