FOUNDATIONS OF SET THEORY
My second reservation about the indispensability arguments rests on somewhat less familiar grounds; to reach it, I must review a bit of set theory and (in the next section) a bit of physics.
It is well-known that the standard axioms of contemporary set theory, ZFC, are not enough to decide every naturally-arising set theoretical question.’6 The most famous independent statement is the Cantor’s continuum hypothesis (CH), but there are others, some of them more down-to-earth than CH. For example, between the mid-seventeenth and the late nineteenth century, under pressure from both physical and mathematical problems, the notion of func-
1 “Reply to Charles Parsons,” p. 400. 15 This issue of the continuity of space-time will take an unexpected turn in
sect. V. 16 Kurt Godel’s incompleteness theorem is enough to establish that there are
set-theoretic statements that can neither be proved nor disproved from ZFC, but Godel’s later work on the inner model of constructible sets and Paul Cohen’s forcing methods yield more: there are statements which mathematicians have found it natural to ask which are likewise independent.
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tion became more and more general. Around the beginning of our century, various mathematicians undertook to bring order to the wild domain of discontinuous functions. It soon became clear that the complexity of functions could be understood in terms of the complexity of sets of real numbers-e.g., a function is continuous if the inverse image of every open set is open-and this naturally led to a serious study of the properties of definable sets of reals. Among these, the Borel sets could easily be shown, for example, to be Lebesgue measurable. This result generalized to projections’7 of Borel sets (the analytic sets) and to the complements of these (the coanalytic sets). One more application of projection produces the sets we now call :2, but the question of their Lebesgue measurabi- lity remained stubbornly unsolved. Sometime later, this question was shown to be independent of ZFC.’8
In contrast with CH, this question concerns only a limited class of definable sets of reals, sets whose definitions have concrete geomet- ric interpretations, and it involves the intuitive analytic notion of Lebesgue measurability rather than Cantor’s bold new invention, the comparison of infinite cardinalities. In other words, it might be said that this independent question, unlike CH, arose in the straightforward pursuit of analysis-as-usual. And there are others of this type.
There is a serious foundational debate about the status of these statements. Despite their independence of ZFC, one might hold that there is nevertheless a fact of the matter, that the statements are nevertheless either true or false, and that it is the burden of further theorizing to determine which.’9 At the other extreme, another might insist that ZFC is all there is to set theory, that a statement independent of these axioms has no inherent truth value, that the study of extensions of ZFC that settle these questions one way or the other are all equally legitimate. For future reference, let me attach labels to crude versions of these positions: letfact be the bare as- sumption that there is a determinate answer to our question, and let the opposing view be no-fact.
Now let us pose this foundational question to the indispensability theorist, taking the simple version of indispensability first, for pur-
17 The projection of a subset of the plane is its shadow on one of the coordinate axes.
1 I discuss this history in more detail in ch. 4 of Realism in Mathematics and in “Taking Naturalism Seriously.”
19 This was Godel’s view, and the one defended in Realism in Mathematics.
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poses of comparison. Continuum mathematics, including every- thing from real-valued measurement to the higher calculus, is among the most widely applied branches of mathematics, and at least some of the many physical theories in which it is applied are extremely well-confirmed. Therefore, so the argument goes, we have good reason to believe in the entities of continuum mathemat- ics, for example, the real numbers. In these applications-for exam- ple, in the theory of space (or space-time)-we also find quantifica- tion over sets of reals, (or equivalently, over regions of space (or space-time)), though particular instances are rarely as complex as
1 .21 If we believe in the reals and in those sets of reals definable in our theory, then it seems we should accept the legitimacy of the question: Are ‘ sets Lebesgue measurable?22 Thus the simple indis- pensability theorist should endorse fact.
What follows from this for the practice of set theory? Should the set theorist, as Godel suggests, seek an answer to this legitimate question? Given that our independent question seems, for now, to have no bearing on physical theory, and that it is not settled by the most generous of “simplificatory rounding outs” (i.e., ZFC), the simple indispensability theorist, who uses only the justificatory methods of physical science, has no means for answering it. Further- more, the exclusive focus on the needs and methods of physical science hints at a lack of interest in any question without physical ramifications. If so, the simple theorist may disagree with no-fact, classifying our independent question as one with an unambiguous truth value, without going so far as charging future theorists with the task of answering it. Call this weak fact.
Fortunately, these subtle matters of interpretation are beside the point here, because we have already identified the modified indis- pensability argument as more promising than the simple. Like the simple theorist, the modified indispensability theorist embraces the ontology of continuum mathematics, on the basis of its successful applicability, and thus the legitimacy of our independent question, but she goes beyond the simple theorist by ratifying the set theorist’s
20 For now, I shall ignore the scientific practice objection and take fundamental scientific theory at face value.
21 One theoretical proposal that involves nonmeasurable sets of reals is I. Pi- towsky, “Deterministic Model of Spin and Statistics,” Physical Review D, xxvii, 10 (15 May 1983): 2316-26. (M. van Lambalgen called this paper to my atten- tion.)
22 This might be avoided if we took mathematical entities to be somehow “in- complete,” an idea of Parsons’s which Quine considers in passing. See Quine’s “Reply to Charles Parsons,” p. 401, and references cited there.
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search for new axioms to answer the question. Call this strong fact. Although the evidence for or against these axiom candidates will derive, not from physical applications, but from considerations in- ternal to mathematics, the modified theorist sees the past success of such mathematical methods as justifying their contin- ued use.
Either way, then, the indispensability theorist should adopt some version of fact. Notice, however, that this acceptance of the legiti- macy of our independent question and (for the modified theorist) the legitimacy of its pursuit is not unconditional; it depends on the empirical facts of current science. The resulting mathematical be- liefs are likewise a posteriori and fallible.