Philosophical Perspectives, 17, Language and Philosophical Linguistics, 2003
ALL SETS GREAT AND SMALL: AND I DO MEAN AL...

Author:
Shapiro Stewart

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Philosophical Perspectives, 17, Language and Philosophical Linguistics, 2003

ALL SETS GREAT AND SMALL: AND I DO MEAN ALL

Stewart Shapiro The Ohio State University The University of St. Andrews I want it all. Queen You can’t always get what you want. But if you try sometime, you just might find, you get what you need. Rolling Stones

1. Wither generality? Timothy Williamson [2003] has made a compelling, prima facie case against the view he calls ‘‘generality relativism’’, the thesis that it is not possible for firstorder variables to range over everything at once. He and others have pointed out that one cannot state the relativist position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of the word ‘‘something’’ at the end? Or suppose we ask the relativist if there is some one thing cannot appear in the range of any bound variable. The likely response would be: ‘‘No. For each object o, it possible to include o in the range of quantifiers, but one cannot quantify over everything at once.’’ This sentence contains an unrestricted quantifier, or so it seems, pending some clever move from a generality relativist. Truth be told, I am not particularly interested in whether it is coherent to have bound variables ranging over absolutely everything. When it comes to the world of subatomic physics, for example, who knows if it is best to talk about objects at all, let alone all objects? The same may go for ordinary talk of items with vague boundaries, such as clouds, mountains, and seas. In response to this, I can hear a Quinean generalist protesting.1 The trouble, if there is trouble, lies with predicates like ‘‘particle’’ and ‘‘cloud’’, not with ‘‘exists’’ or, what is the same thing, with quantification. If there are such things as particles and clouds, then they fall within the range of our bound variables, and can do so all at once. However, I am not sure matters are this straightforward. On some views of

468 / Stewart Shapiro

vagueness, the boundaries of vague terms vary with context. Any context can change and, with this change, new objects might be found within the range of the quantifiers. It may not make sense to have a super-context that includes every context. These matters of anti-realism and metaphysics can be put aside here. I am concerned with the special case of whether one can have bound variables ranging over all pure sets, or all pure set-like-totalities. And I am interested in whether one can have bound variables ranging over all ordinals and all cardinals, or over all well-ordering-types and cardinality-types. I presume that those are the interesting cases anyway, given the role of the Russell, Cantor, and Burali-Forti paradoxes in the literature on this topic. Toward the end of his paper, Williamson puts his finger on what I take to be the main semi-formal sticking point for the generality absolutist: how are we to understand second-order quantifiers when the corresponding first-order quantifiers have unrestricted range? Prima facie, second-order quantifiers have a range too, and by Russell’s paradox, this range cannot lie entirely in the range of the firstorder variables. So there seems to be something that cannot lie in the range of first-order quantifiers. So first-order quantifiers cannot be completely unrestricted. The same goes for any quantifier at any level, provided that there is a higher level. Mea culpa. In my book on second-order logic (Shapiro [1991]), I took the second-order variables of set theory to range over proper classes, which I called ‘‘logical sets’’. The lesson of Russell’s paradox, I said, is that in the context of set theory, there are logical sets that do not correspond to any member of the iterative hierarchy. In the meta-theory, or perhaps the meta-meta-theory, or the mathematical English that I used to write the book, I suppose I took classes to be ‘‘things’’. Along with many (but not all) set-theorists, I used singular terms, like ‘‘V’’ and ‘‘V’’, that denote proper classes, and I had informal meta-variables ranging over proper classes. Clearly, proper classes are set-like things, having only pure iterative sets as members. So the first-order variables of second-order ZFC do not range over all pure set-like things. But consider George Boolos’s [1998, 35] retort to a similar suggestion: ‘‘Wait a minute! I thought that set theory was supposed to be a theory about all, ‘absolutely’ all, the collections that there were and that ‘set’ was synonymous with ‘collection’.’’2 At the time, I might have responded with the above line from the Rolling Stones: ‘‘You can’t always get what you want’’, perhaps adding that we can get what we need. I see now, if not then, that there is something fishy about claiming that second-order ZFC is the most inclusive theory of pure sets that there is, and then using informal variables ranging over pure set-like things that outrun the first-order variables of ZFC on its intended interpretation. To solve this problem, Boolos proposed his celebrated plural interpretation of monadic, second-order quantifiers. Williamson raises five points against that resolution, at least four of which apply in the cases that interest me: pure sets, ordinals, and cardinals. I have my own doubts as to whether our independent or pre-theoretic grasp of plural quantifiers is sufficiently determinate to ground

All Sets Great and Small / 469

second-order theories with infinite, let alone unbounded, domains. Consider a statement of second-order real analysis of the form: "X9Y(X,Y).

There is no issue concerning the existence of sets of real numbers, or at least none that is relevant here (and none that moved Boolos himself). So the opening second-order quantifiers can be given both a plural and an ordinary, set-theoretic interpretation. It had better be the case that if we read the quantifiers as plurals, we will get exactly the same truth value, in general, as we would if we understand the quantifiers as ranging over sets of real numbers. In effect, there needs to be a ‘‘plurality’’ (if you will excuse the expression) for each set of real numbers. Does the English plural construction have that determinate a meaning? Of course, the pluralist can always stipulate that she intends it to have such a meaning in cases like that of real analysis—but one needs some set theory to make the stipulation. This move might sustain Michael Resnik’s [1988] and my [1993] complaint that the sophisticated understanding of the plural construction used in justifying second-order logic is mediated by set theory. In any case, the important use of the plural construction is for cases where we would rather not—or cannot—speak of pluralities as things. Second-order Zermelo-Fraenkel set theory is the main case in point (as confirmed in conversation with Boolos). Is there reason to think that the plural construction is sufficiently determinate in such cases? Is there a ‘‘plurality’’ corresponding to each and every proper class? The pluralist is surely not in position to stipulate this, unless she recognizes proper classes as objects. I do not claim to have presented a knock-down objection against the plural rescue. Moreover, there are a number of other proposals for higher-order quantification that need to be digested. Williamson suggests that we can directly engender an understanding of the generality involved in second-order quantification, an understanding that is not mediated by set theory or any other construal of the range of the quantifiers. Nevertheless, this direct understanding is supposedly equivalent to the set-theoretic interpretation on domains in which the domain is a set (with a powerset). If the Williamson plan succeeds, and does not beg any questions, then we can wax homophonic in giving truth conditions. Perhaps. I take it as agreed that the antinomies provide the basic motivation for generality relativism. If the absolutist gets past those, his remaining problematic feature is the use of unrestricted second-order quantification. However, even if first-order languages are too weak, we may not need full second-order languages to do whatever work we want our grand theory to do. Here, I float a compromise, building on the framework proposed in Zermelo [1930], ‘‘U¨ber Grenzzahlen und Mengenbereiche’’ (‘‘On boundary numbers and domains of sets’’). It allows unrestricted, absolute first-order quantification, and it allows at least restricted second-order quantification. To echo the Rolling Stones, extendibility principles give us what we need. Although similar frameworks have been proposed by

470 / Stewart Shapiro

Charles Parsons and Geoffrey Hellman, they are generality-relativists, after a fashion. Let’s see how much of the cake we can have if we eat it too. We turn first to one of the antinomies.

2. Pesky Burali-Forti and indefinite extensibility Russell’s ‘‘On some difficulties in the theory of transfinite numbers and order types’’ [1906] begins with an examination of the now standard paradoxes, and concludes: …the contradictions result from the fact that…there are what we may call selfreproductive processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new term also having the property in question. Hence we can never collect all of the terms having the said property into a whole; because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property.

Citing this passage, Michael Dummett [1993, 441] writes that an ‘‘indefinitely extensible concept is one such that, if we can form a definite conception of a totality all of whose members fall under the concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it’’ (emphasis mine). According to Dummett, an indefinitely extensible property P has a ‘‘principle of extension’’ that takes any definite totality t of objects each of which has P, and produces an object that also has P, but is not in t (see also Dummett [1991, 316–319]). Let us say that a property P is Definite if it is not indefinitely extensible. Dummett’s remarks won’t do as a definition, since he uses the word ‘‘definite’’ to explain what it is to be indefinitely extensible. Nevertheless, what he says seems correct. Let us focus on the Burali-Forti paradox. Let O be any Definite collection of ordinal numbers. Let O0 be the collection of all ordinals a such that there is a b2O and ab. That is, a is in O0 if a is smaller than, or equal to, something in O. Since O0 is well-ordered, let g be its order type. Let g0 be the order type of O0 [ {g}—the order of O0 with one item added ‘‘at the end’’. Then g0 is an ordinal number, and g0 is not a member of O. So the property of being an ordinal is indefinitely extensible. One can, of course, challenge the set-theoretic principles—union, pairing, etc.—used here, but the reasoning does appear natural. As Dummett [1991, 316] puts it, if we have a clear grasp of any totality of ordinals, we thereby have a conception of what is intuitively an ordinal number greater than any member of that totality. Any [D]efinite totality of ordinals must therefore be so circumscribed

All Sets Great and Small / 471 as to forswear comprehensiveness, renouncing any claim to cover all that we might intuitively recognise as being an ordinal.

This is the sort of thing that motivates generality relativism. Russell [1906, 144] wrote that it ‘‘is probable’’ that if P is any property which demonstrably does not have an extension (that obeys extensionality) then ‘‘we can actually construct a series, ordinally similar to the series of all ordinals, composed entirely of terms having the property’’ P. In present terms, Russell’s conjecture is that if P is indefinitely extensible, then there is a one-to-one function from the ordinals into P. Russell does not provide an argument for this, but I think there is one: Let a be an ordinal and assume that we have a one-to-one function f from the ordinals smaller than a to objects that have the property P. Consider the collection {fb b < a}. This is Definite. Since P is indefinitely extensible, there is an object a such that P holds of a, but a is not in this set. Set fa = a.

This argument uses transfinite recursion on ordinals and a version of replacement: if a totality t is equinumerous with an ordinal, then t is Definite.3 Both of those seem beyond reproach, but one does need special care in areas like this. It is clear that some intuitive principles have to be dropped. Nevertheless, if the argument (or at least its conclusion) is correct, then ‘‘ordinal’’ is the basic indefinitely extensible notion. In any case, the Burali-Forti paradox is robust. The very definition of well-ordering suffices to generate ever more ordinals, or at least what look like well-ordering types, without using ‘‘external’’ resources like the powerset or setcomprehension principles invoked in the Cantor and Russell paradoxes. Let V(x) be the property of being an ordinal. It is, of course, routine to show that V is itself a well-ordering (i.e., has the requisite property of relations). That is, the V’s are well-ordered. But, alas, V has no order type. We are used to that. We can define a relation that is a well-ordering strictly longer than V: Let a and b be ordinals. Say that a1 b if a6¼0 and either a

ALL SETS GREAT AND SMALL: AND I DO MEAN ALL

Stewart Shapiro The Ohio State University The University of St. Andrews I want it all. Queen You can’t always get what you want. But if you try sometime, you just might find, you get what you need. Rolling Stones

1. Wither generality? Timothy Williamson [2003] has made a compelling, prima facie case against the view he calls ‘‘generality relativism’’, the thesis that it is not possible for firstorder variables to range over everything at once. He and others have pointed out that one cannot state the relativist position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of the word ‘‘something’’ at the end? Or suppose we ask the relativist if there is some one thing cannot appear in the range of any bound variable. The likely response would be: ‘‘No. For each object o, it possible to include o in the range of quantifiers, but one cannot quantify over everything at once.’’ This sentence contains an unrestricted quantifier, or so it seems, pending some clever move from a generality relativist. Truth be told, I am not particularly interested in whether it is coherent to have bound variables ranging over absolutely everything. When it comes to the world of subatomic physics, for example, who knows if it is best to talk about objects at all, let alone all objects? The same may go for ordinary talk of items with vague boundaries, such as clouds, mountains, and seas. In response to this, I can hear a Quinean generalist protesting.1 The trouble, if there is trouble, lies with predicates like ‘‘particle’’ and ‘‘cloud’’, not with ‘‘exists’’ or, what is the same thing, with quantification. If there are such things as particles and clouds, then they fall within the range of our bound variables, and can do so all at once. However, I am not sure matters are this straightforward. On some views of

468 / Stewart Shapiro

vagueness, the boundaries of vague terms vary with context. Any context can change and, with this change, new objects might be found within the range of the quantifiers. It may not make sense to have a super-context that includes every context. These matters of anti-realism and metaphysics can be put aside here. I am concerned with the special case of whether one can have bound variables ranging over all pure sets, or all pure set-like-totalities. And I am interested in whether one can have bound variables ranging over all ordinals and all cardinals, or over all well-ordering-types and cardinality-types. I presume that those are the interesting cases anyway, given the role of the Russell, Cantor, and Burali-Forti paradoxes in the literature on this topic. Toward the end of his paper, Williamson puts his finger on what I take to be the main semi-formal sticking point for the generality absolutist: how are we to understand second-order quantifiers when the corresponding first-order quantifiers have unrestricted range? Prima facie, second-order quantifiers have a range too, and by Russell’s paradox, this range cannot lie entirely in the range of the firstorder variables. So there seems to be something that cannot lie in the range of first-order quantifiers. So first-order quantifiers cannot be completely unrestricted. The same goes for any quantifier at any level, provided that there is a higher level. Mea culpa. In my book on second-order logic (Shapiro [1991]), I took the second-order variables of set theory to range over proper classes, which I called ‘‘logical sets’’. The lesson of Russell’s paradox, I said, is that in the context of set theory, there are logical sets that do not correspond to any member of the iterative hierarchy. In the meta-theory, or perhaps the meta-meta-theory, or the mathematical English that I used to write the book, I suppose I took classes to be ‘‘things’’. Along with many (but not all) set-theorists, I used singular terms, like ‘‘V’’ and ‘‘V’’, that denote proper classes, and I had informal meta-variables ranging over proper classes. Clearly, proper classes are set-like things, having only pure iterative sets as members. So the first-order variables of second-order ZFC do not range over all pure set-like things. But consider George Boolos’s [1998, 35] retort to a similar suggestion: ‘‘Wait a minute! I thought that set theory was supposed to be a theory about all, ‘absolutely’ all, the collections that there were and that ‘set’ was synonymous with ‘collection’.’’2 At the time, I might have responded with the above line from the Rolling Stones: ‘‘You can’t always get what you want’’, perhaps adding that we can get what we need. I see now, if not then, that there is something fishy about claiming that second-order ZFC is the most inclusive theory of pure sets that there is, and then using informal variables ranging over pure set-like things that outrun the first-order variables of ZFC on its intended interpretation. To solve this problem, Boolos proposed his celebrated plural interpretation of monadic, second-order quantifiers. Williamson raises five points against that resolution, at least four of which apply in the cases that interest me: pure sets, ordinals, and cardinals. I have my own doubts as to whether our independent or pre-theoretic grasp of plural quantifiers is sufficiently determinate to ground

All Sets Great and Small / 469

second-order theories with infinite, let alone unbounded, domains. Consider a statement of second-order real analysis of the form: "X9Y(X,Y).

There is no issue concerning the existence of sets of real numbers, or at least none that is relevant here (and none that moved Boolos himself). So the opening second-order quantifiers can be given both a plural and an ordinary, set-theoretic interpretation. It had better be the case that if we read the quantifiers as plurals, we will get exactly the same truth value, in general, as we would if we understand the quantifiers as ranging over sets of real numbers. In effect, there needs to be a ‘‘plurality’’ (if you will excuse the expression) for each set of real numbers. Does the English plural construction have that determinate a meaning? Of course, the pluralist can always stipulate that she intends it to have such a meaning in cases like that of real analysis—but one needs some set theory to make the stipulation. This move might sustain Michael Resnik’s [1988] and my [1993] complaint that the sophisticated understanding of the plural construction used in justifying second-order logic is mediated by set theory. In any case, the important use of the plural construction is for cases where we would rather not—or cannot—speak of pluralities as things. Second-order Zermelo-Fraenkel set theory is the main case in point (as confirmed in conversation with Boolos). Is there reason to think that the plural construction is sufficiently determinate in such cases? Is there a ‘‘plurality’’ corresponding to each and every proper class? The pluralist is surely not in position to stipulate this, unless she recognizes proper classes as objects. I do not claim to have presented a knock-down objection against the plural rescue. Moreover, there are a number of other proposals for higher-order quantification that need to be digested. Williamson suggests that we can directly engender an understanding of the generality involved in second-order quantification, an understanding that is not mediated by set theory or any other construal of the range of the quantifiers. Nevertheless, this direct understanding is supposedly equivalent to the set-theoretic interpretation on domains in which the domain is a set (with a powerset). If the Williamson plan succeeds, and does not beg any questions, then we can wax homophonic in giving truth conditions. Perhaps. I take it as agreed that the antinomies provide the basic motivation for generality relativism. If the absolutist gets past those, his remaining problematic feature is the use of unrestricted second-order quantification. However, even if first-order languages are too weak, we may not need full second-order languages to do whatever work we want our grand theory to do. Here, I float a compromise, building on the framework proposed in Zermelo [1930], ‘‘U¨ber Grenzzahlen und Mengenbereiche’’ (‘‘On boundary numbers and domains of sets’’). It allows unrestricted, absolute first-order quantification, and it allows at least restricted second-order quantification. To echo the Rolling Stones, extendibility principles give us what we need. Although similar frameworks have been proposed by

470 / Stewart Shapiro

Charles Parsons and Geoffrey Hellman, they are generality-relativists, after a fashion. Let’s see how much of the cake we can have if we eat it too. We turn first to one of the antinomies.

2. Pesky Burali-Forti and indefinite extensibility Russell’s ‘‘On some difficulties in the theory of transfinite numbers and order types’’ [1906] begins with an examination of the now standard paradoxes, and concludes: …the contradictions result from the fact that…there are what we may call selfreproductive processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new term also having the property in question. Hence we can never collect all of the terms having the said property into a whole; because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property.

Citing this passage, Michael Dummett [1993, 441] writes that an ‘‘indefinitely extensible concept is one such that, if we can form a definite conception of a totality all of whose members fall under the concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it’’ (emphasis mine). According to Dummett, an indefinitely extensible property P has a ‘‘principle of extension’’ that takes any definite totality t of objects each of which has P, and produces an object that also has P, but is not in t (see also Dummett [1991, 316–319]). Let us say that a property P is Definite if it is not indefinitely extensible. Dummett’s remarks won’t do as a definition, since he uses the word ‘‘definite’’ to explain what it is to be indefinitely extensible. Nevertheless, what he says seems correct. Let us focus on the Burali-Forti paradox. Let O be any Definite collection of ordinal numbers. Let O0 be the collection of all ordinals a such that there is a b2O and ab. That is, a is in O0 if a is smaller than, or equal to, something in O. Since O0 is well-ordered, let g be its order type. Let g0 be the order type of O0 [ {g}—the order of O0 with one item added ‘‘at the end’’. Then g0 is an ordinal number, and g0 is not a member of O. So the property of being an ordinal is indefinitely extensible. One can, of course, challenge the set-theoretic principles—union, pairing, etc.—used here, but the reasoning does appear natural. As Dummett [1991, 316] puts it, if we have a clear grasp of any totality of ordinals, we thereby have a conception of what is intuitively an ordinal number greater than any member of that totality. Any [D]efinite totality of ordinals must therefore be so circumscribed

All Sets Great and Small / 471 as to forswear comprehensiveness, renouncing any claim to cover all that we might intuitively recognise as being an ordinal.

This is the sort of thing that motivates generality relativism. Russell [1906, 144] wrote that it ‘‘is probable’’ that if P is any property which demonstrably does not have an extension (that obeys extensionality) then ‘‘we can actually construct a series, ordinally similar to the series of all ordinals, composed entirely of terms having the property’’ P. In present terms, Russell’s conjecture is that if P is indefinitely extensible, then there is a one-to-one function from the ordinals into P. Russell does not provide an argument for this, but I think there is one: Let a be an ordinal and assume that we have a one-to-one function f from the ordinals smaller than a to objects that have the property P. Consider the collection {fb b < a}. This is Definite. Since P is indefinitely extensible, there is an object a such that P holds of a, but a is not in this set. Set fa = a.

This argument uses transfinite recursion on ordinals and a version of replacement: if a totality t is equinumerous with an ordinal, then t is Definite.3 Both of those seem beyond reproach, but one does need special care in areas like this. It is clear that some intuitive principles have to be dropped. Nevertheless, if the argument (or at least its conclusion) is correct, then ‘‘ordinal’’ is the basic indefinitely extensible notion. In any case, the Burali-Forti paradox is robust. The very definition of well-ordering suffices to generate ever more ordinals, or at least what look like well-ordering types, without using ‘‘external’’ resources like the powerset or setcomprehension principles invoked in the Cantor and Russell paradoxes. Let V(x) be the property of being an ordinal. It is, of course, routine to show that V is itself a well-ordering (i.e., has the requisite property of relations). That is, the V’s are well-ordered. But, alas, V has no order type. We are used to that. We can define a relation that is a well-ordering strictly longer than V: Let a and b be ordinals. Say that a1 b if a6¼0 and either a

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close