7 Sets and Hypersets

0465037704-02.qxd 8/23/00 9:51 AM Page 140

Some Conceptual Issues in Set Theory We have been looking at mathematics from a cognitive perspective, seeking to understand the cognitive and ultimately the bodily basis of mathematics. Set theory is at the heart of modern mathematics. It is therefore incumbent upon us to understand the ideas implicit in set theory as well as we can. In our discussion of Boolean classes in Chapter 6, we saw that the concept of a class is experientially grounded via the metaphor that Classes Are Contain- ers—mental containers that we use in perceiving, conceptualizing, and reason- ing about our experience. As we saw there, notions like the empty class and the technical notion of subclass (in which a class is a subclass of itself) are intro- duced via Boole’s metaphor. Modern set theory begins with these basic elements of our grounding metaphor for classes and with Boole’s metaphor. That is, it starts with the notion of a class as a containerlike entity; the ideas of intersection, union, and complement; the associative, commutative, and distributive laws; the empty set; and the idea of closure. Sets are more sophisticated than Boolean classes. Take a simple case. In Boolean algebra, classes can be subclasses of other classes but not members of those classes. Contemporary set theory allows sets to be members, not just subclasses, of other sets. What is the grounding for the intuitive conceptual distinction between sets as subsets and sets as members? As subsets, intuitive sets are Container schemas, mental containers organizing objects into groups. For sets to be members in this conceptualization, they, too, must be conceptualized as objects on a par with other objects and not just mental Container schemas providing organizations for objects. To conceptualize

140 0465037704-02.qxd 8/23/00 9:51 AM Page 141

Sets and Hypersets 141

sets as objects, we need a conceptual metaphor providing such a conceptualization. We will call this metaphor Sets Are Objects. Via this metaphor, the empty set can be viewed not just as a subset but as a unique entity in its own right—an entity that may or may not be a member of a given set, even though it is a subset of every set. The Sets Are Objects metaphor allows us to conceptualize power sets, in which all the subsets of a given set A are made into the members of another set, P(A). For example, consider the set A = {a, b, c}. The following are the subsets of A: , {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}. The power set, P(A) has these subsets of A as its members. P(A) = {, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c} }. Note that the set {a, b, c} is a member of its power set. Incidentally, as long as we conceptualize sets as Container schemas, it will be impossible to conceptualize sets as members of themselves. The reason is that a member is properly included in the interior of a Container schema, and no Container schema can be in its own interior.

The Ordered Pair Metaphor Once we conceptualize sets as objects that can be members of another set, we can construct a metaphorical deﬁnition for ordered pairs. Intuitively, an ordered pair is conceptualized nonmetaphorically as a subitized pair of elements (by what we will call a Pair schema) structured by a Path schema, where the source of the path is seen as the ﬁrst member of the pair and the goal of the path is seen as the second member. This is simply our intuitive notion of what an ordered pair is. With the addition of the Sets Are Objects metaphor, we can conceptualize ordered pairs metaphorically, not in terms of Path and Pair schemas but in terms of sets:

The Ordered Pair Metaphor Source Domain Target Domain Sets Ordered Pairs The Set {{a}, {a, b}} Æ The Ordered Pair (a, b)

Using this metaphorical concept of an ordered pair, one can go on to metaphorically deﬁne relations, functions, and so on in terms of sets. One of the most interesting things that one can do with this metaphorical ordered-pair deﬁnition is to metaphorically conceptualize the natural numbers in terms of sets that have other sets as members, as the great mathematician John von Neumann (1903–1957) did. 0465037704-02.qxd 8/23/00 9:51 AM Page 142

142 Where Mathematics Comes From

The Natural Numbers Are Sets Metaphor Source Domain Target Domain Sets Natural Numbers The empty set Æ Zero The set containing the Æ One empty set {} (i.e., {0}). The set {, {}} (that is, {0, 1}) Æ Two The set {, {}, {, {}}} (i.e., {0, 1, 2}) Æ Three The set of its predecessors Æ A natural number (built following the above rule)

Here, the set with no members is mapped onto the number zero, the set containing the empty set as a member (i.e., it has one member) is mapped onto the number one, the set with two members is mapped onto the number two, and so on. By virtue of this metaphor, every set containing three members is in a one- to-one correspondence with the number three. Using this metaphor, you can metaphorically construct the natural numbers out of nothing but sets. From a cognitive perspective, this is a metaphor that allows us to conceptualize numbers, which are one kind of conceptual entity, in terms of sets, which are a very different kind of conceptual entity. This is a linking metaphor—a metaphor that allows one to conceptualize one branch of mathematics (arithmetic) in terms of another branch (set theory). Linking metaphors are different from grounding metaphors in that both the source and target domains of the mapping are within mathematics itself. Such linking metaphors, where the source domain is set theory, are what is needed conceptually to “reduce” other branches of mathematics to set theory in order to satisfy the Formal Founda- tions metaphor. (For a discussion of the Formal Foundations program and the metaphors needed to achieve it, see Chapter 16.)

Cantor’s Metaphor Our ordinary everyday conceptual system includes the concepts Same Number As and More Than. They are based, of course, on our experience with ﬁnite, not inﬁnite, collections. Among the criteria characterizing our ordinary everyday versions of these concepts are:

• Same Number As: Group A has the same number of elements as group B if, for every member of A, you can take away a corresponding member of B and not have any members of B left over. 0465037704-02.qxd 8/23/00 9:51 AM Page 143

Sets and Hypersets 143

• More Than: Group B has more objects than group A if, for every member of A, you can take away a member of B and still have members left in B.

It is important to contrast our everyday concept of Same Number As with the German mathematician Georg Cantor’s concept of pairability—that is, capable of being put into a one-to-one correspondence. Finite collections with the same number of objects are pairable, since two finite sets with the same number of objects can be put in a one-to-one correspondence. This does not mean that Same Number As and pairability are the same idea; the ideas are different in a significant way, but they happen to correlate precisely for finite sets. The same is not true for infinite sets. Compare the set of natural numbers and the set of even integers. As Cantor (1845–1918) observed, they are pairable; that is, they can be put into a one-to- one correspondence. Just multiply the natural numbers by two, and you will set up the one-to-one correspondence (see Figure 7.1). Of course, these two sets do not have the same number of elements according to our everyday criterion. If you take the even numbers away from the natural numbers, there are still all the odd numbers left over. According to our usual concept of “more than,” there are more natural numbers than even numbers, so the concepts Same Number As and pairability are different concepts. Our everyday concepts of Same Number As and More Than are, of course, linked to other everyday quantitative concepts, like How Many or As Many As, and Size Of, as well as to the concept Number itself—the basic concept in arithmetic. In his investigations into the properties of infinite sets, Cantor used the concept of pairability in place of our everyday concept of Same Number As. In doing so, he established a conceptual metaphor, in which one concept (Same Number As) is conceptualized in terms of the other (pairability).

Cantor’s Metaphor Source Domain Target Domain Mappings Numeration Set A and set B can be put into Set A and set B have the same Æ one-to-one correspondence number of elements

That is, our ordinary concept of having the same number of elements is metaphorically conceptualized, especially for inﬁnite sets, in terms of the very different concept of being able to be put in a one-to-one correspondence. This distinction has never before been stated explicitly using the idea of conceptual metaphor. Indeed, because the distinction has been blurred, generations of students have been confused. Consider the following statement made by 0465037704-02.qxd 8/23/00 9:51 AM Page 144

144 Where Mathematics Comes From

FIGURE 7.1 The one-to-one correspondence between the natural numbers and the positive even numbers.

many mathematics teachers: “Cantor proved that there are just as many positive even integers as natural numbers.” Given our ordinary concept of “As Many As,” Cantor proved no such thing. He proved only that the sets were pairable. In our ordinary conceptual system, there are more natural numbers than there are positive even integers. It is only by use of Cantor’s metaphor that it is correct to say that he proved that there are, metaphorically, “just as many” even numbers as natural numbers. The same comment holds for other proofs of Cantor’s. According to our ordinary concept of “More Than,” there are more rational numbers than natural numbers, since if you take the natural numbers away from the rational numbers, there will be lots left over. But Cantor did prove that the two sets are pairable, and hence they can be said (via Cantor’s metaphor) to metaphorically have the “same number” of elements. (For details, see Chapter 10.) Cantor’s invention of the concept of pairability and his application of it to infinite sets was a great conceptual achievement in mathematics. What he did in the process was create a new technical mathematical concept—pairability—and with it, new mathematics. But Cantor also intended pairability to be a literal extension of our ordinary notion of Same Number As from finite to infinite sets. There Cantor was mistaken. From a cognitive perspective, it is a metaphorical rather than literal extension of our everyday concept. The failure to teach the difference between Cantor’s technical metaphorical concept and our ordinary concept confuses generation after generation of introductory students.

Axiomatic Set Theory and Hypersets Axiomatic Set Theory In Chapter 16, we will discuss the approach to mathematics via formal systems of axioms. We will see there that such an approach requires a linking metaphor for conceptualizing complex and sophisticated mathematical ideas in terms of much simpler mathematical ideas, those of set theory and the theory of formal 0465037704-02.qxd 8/23/00 9:51 AM Page 145

Sets and Hypersets 145

systems. For example, there must be a metaphor for characterizing what it means for a mathematical structure to “fit” a collection of axioms that have no inherent meaning. Whereas Euclid understood his postulates for geometry to be meaningful to human beings, formal systems of axioms are taken as mind-free sequences of symbols. They are defined to be free of human conceptual systems and human understanding. On the formalist view of the axiomatic method, a “set” is any mathematical structure that “satisfies” the axioms of set theory as written in symbols. The traditional axioms for set theory (the Zermelo-Fraenkel axioms) are often taught as being about sets conceptualized as containers. Many writers speak of sets as “containing” their members, and most students think of sets that way. Even the choice of the word “member” suggests such a reading, as do the Venn diagrams used to introduce the subject. But if you look carefully through those axioms, you will find nothing in them that characterizes a container. The terms “set” and “member of” are both taken as undefined primitives. In formal mathematics, that means that they can be anything that fits the axioms. Here are the classic Zermelo-Fraenkel axioms, including the axiom of choice; together they are commonly called the ZFC axioms.

• The axiom of Extension: Two sets are equal if and only if they have the same members. In other words, a set is uniquely determined by its members. • The axiom of Speciﬁcation: Given a set A and a one-place predicate P(x) that is either true or false of each member of A, there exists a subset of A whose members are exactly those members of A for which P(x) is true. • The axiom of Pairing: For any two sets, there exists a set that they are both members of. • The axiom of Union: For every collection of sets, there is a set whose members are exactly the members of the sets of that collection. • The axiom of Powers: For each set A, there is a set P(A) whose members are exactly the subsets of set A. • The axiom of Inﬁnity: There exists a set A such that (1) the empty set is a member of A, and (2) if X is a member of A, then the successor of X is a member of A. • The axiom of Choice: Given a disjoint set S whose members are non- empty sets, there exists a set C which has as its members one and only one element from each member of S.

There is nothing in these axioms that explicitly requires sets to be containers. What these axioms do, collectively, is to create entities called “sets,” ﬁrst 0465037704-02.qxd 8/23/00 9:51 AM Page 146

146 Where Mathematics Comes From

from elements and then from previously created classes and sets. The axioms do not say explicitly how sets are to be conceptualized. The point here is that, within formal mathematics, where all mathematical concepts are mapped onto set-theoretical structures, the “sets” used in these structures are not technically conceptualized as Container schemas. They do not have Container schema structure with an interior, boundary, and exterior. Indeed, within formal mathematics, there are no concepts at all, and hence sets are not conceptualized as anything in particular. They are undefined entities whose only constraints are that they must “fit” the axioms. For formal logi- cians and model theorists, sets are those entities that fit the axioms and are used in the modeling of other branches of mathematics. Of course, most of us do conceptualize sets in terms of Container schemas, and that is perfectly consistent with the axioms given above. However, when we conceptualize sets as Container schemas, a constraint follows automati- cally: Sets cannot be members of themselves, since containers cannot be inside themselves. Strictly speaking, this constraint does not follow from the axioms but from our metaphorical understanding of sets in terms of containers. The axioms do not rule out sets that contain themselves. However, an extra axiom was proposed by von Neumann that does rule out this possibility.

• The axiom of Foundation: There are no inﬁnite descending sequences

of sets under the membership relation; that is, . . . Si+1 Œ Si Œ . . . Œ S is ruled out.

Since allowing sets to be members of themselves would result in such a se- quence, this axiom has the indirect effect of ruling out self-membership.

Hypersets Technically, within formal mathematics, model theory has nothing to do with everyday understanding. Model theorists do not depend upon our ordinary container-based concept of a set. Indeed, certain model theorists have found that our ordinary grounding metaphor that Classes Are Container schemas gets in the way of modeling kinds of phenomena they want to model, especially recursive phenomena. For example, take expressions like

x = 1 + 1 1 + 1 1 + . . . 0465037704-02.qxd 8/23/00 9:51 AM Page 147

Sets and Hypersets 147

If we observe carefully, we can see that the denominator of the main fraction has in fact the value deﬁned for x itself. In other words, the above expression is equivalent to 1 x = 1 + x Such recursive expressions are common in mathematics and computer science. The possibilities for modeling these expressions using “sets” are ruled out if the only kind of “sets” used in the modeling must be ones that cannot have themselves as members. Set theorists have realized that a new noncontainer metaphor is needed for thinking about sets, and they have explicitly constructed one: hyperset theory (Barwise & Moss, 1991). The idea is to use graphs, not containers, for characterizing sets. The kinds of graphs used are Accessible Pointed Graphs, or APGs. “Pointed” denotes an asymmetric relation between nodes in the graph, indicated visually by an arrow pointing from one node to another—or from one node back to that node itself. “Accessible” means that one node is linked to all other nodes in the graph and can therefore be “accessed” from any other node. From the axiomatic perspective, hyperset theorists have replaced the axiom of Foundation with an Anti-Foundation axiom. From a cognitive point of view, the implicit conceptual metaphor they have used is this:

The Sets Are Graphs Metaphor Source Domain Target Domain Accessible Pointed Graphs Sets An APG Æ The membership structure of a set An arrow Æ The membership relation Nodes that are tails of arrows Æ Sets Decorations on nodes that Æ Members are heads of arrows Classical sets with APGs with no loops Æ the Foundation axiom Hypersets with the APGs with or without loops Æ Anti-Foundation axiom

The effect of this metaphor is to eliminate the notion of containment from the concept of a “set.” The graphs have no notion of containment built into them at all. And containment is not modeled by the graphs. 0465037704-02.qxd 8/23/00 9:51 AM Page 148

148 Where Mathematics Comes From

Graphs that have no loops satisfy the ZFC axioms and the axiom of Founda- tion. They thus work just like sets conceptualized as containers. But graphs that do have loops model sets that can have themselves as members; these graphs do not work like sets that are conceptualized as containers, and they do not satisfy the axiom of Foundation. A hyperset is an APG that may or may not contain loops. Hypersets thus ﬁt not the axiom of Foundation but, rather, another axiom with the opposite intent:

• The Anti-Foundation axiom: Every APG pictures a unique set.

The fact that hypersets satisfy the Zermelo-Fraenkel axioms conﬁrms what we said above: The Zermelo-Fraenkel axioms for set theory—the ones generally accepted in mathematics—do not deﬁne our ordinary concept of a set as a container at all! That is, the axioms of “set theory” are not, and were never meant to be, about what we ordinarily call “sets,” which we conceptualize in terms of containers.

Numbers Are Sets, Which Are Graphs If we adjust the von Neumann metaphor for numbers in terms of sets to the new graph-theoretical models, zero (the empty set) becomes a node with no arrows leading from it. One becomes the graph with two nodes and one arrow leading from one node to the other. Two and three are represented by the graphs in Figures 7.2(a) and 7.2(b), respectively. And where an arrow bends back on itself, as in Figure 7.2(c), we get recursion—the equivalent of an infi- nitely long chain of nodes with arrows not bending back on themselves, as in Figure 7.2(d). Here we see the power of conceptual metaphor in mathematics. Sets, conceptualized in everyday terms as containers, do not have the right properties to model everything needed. So we can now metaphorically redeﬁne “sets” to ex- clude containment by using certain kinds of graphs. The only confusing thing is that this special case of graph theory is still called “set theory” for historical reasons. It is sometimes said that the theory of hypersets is “a set theory in which sets can contain themselves.” But this is misleading, because it is not a theory of “sets” as we ordinarily understand them in terms of containment. The reason that these graph-theoretical objects are called “sets” is a functional one: They play the role in modeling axioms that classical sets with the axiom of Founda- tion used to play. 0465037704-02.qxd 8/23/00 9:51 AM Page 149

Sets and Hypersets 149

FIGURE 7.2 Hypersets: sets conceptualized as graphs, with the empty set as the graph having no arrows leading from it. The set containing the empty set is a graph whose root has one arrow leading to the empty set. The metaphor Numbers Are Sets is taken for granted, with zero being the empty set and each natural number being the set of all its predecessors. Draw- ing (a) depicts the number 2, since it is a graph whose root (the top node) has two arrows (membership relations): one leading to the empty set, or zero (the lower right node), and the other leading to the set containing the empty set, or one (the subgraph whose root is the node at the lower left). Drawing (b) depicts the number 3, since its root has three arrows leading to root nodes for subgraphs for the numbers 2, 1, 0, as depicted in (a). Drawing (c) is a graph rep- resenting a set that is a “member” of itself, under the Sets Are Graphs Metaphor. And drawing (d) is an inﬁnitely long chain of nodes in an inﬁnite graph, which is equivalent to (c). (Adapted from Barwise & Moss, 1991.)

What Are Sets, Really? As we have just seen, mathematics has two quite inconsistent metaphorical conceptions of sets—one in terms of Container schemas and one in terms of graphs. Is one of these conceptions right and the other wrong? There is a perspective from which one might think so—a perspective that says there is only one correct notion of a “set.” There is another perspective, the most common one, which recognizes that these two distinct notions of “set” deﬁne different and mutually inconsistent subject matters, conceptualized via radically different metaphors. This situation is much more common in mathematics than most people realize.

The Roles Played by Conceptual Metaphors In this chapter, as in previous chapters, conceptual metaphors have played sev- eral roles. 0465037704-02.qxd 8/23/00 9:51 AM Page 150

150 Where Mathematics Comes From

• First, there are grounding metaphors—metaphors that ground our understanding of mathematical ideas in terms of everyday experience. Ex- amples include the Classes Are Container schemas and the four grounding metaphors for arithmetic. • Second, there are redeﬁnitional metaphors—metaphors that impose a technical understanding replacing ordinary concepts. Cantor’s metaphor is a case in point. • Third, there are linking metaphors—metaphors within mathematics itself that allow us to conceptualize one mathematical domain in terms of another mathematical domain. Examples include Boole’s metaphor, von Neumann’s Natural Numbers Are Sets metaphor, and the Sets Are Graphs metaphor.

The linking metaphors are in many ways the most interesting of these, since they are part of the fabric of mathematics itself. They occur whenever one branch of mathematics is used to model another, as happens frequently. More- over, linking metaphors are central to the creation not only of new mathematical concepts but often of new branches of mathematics. As we shall see, such classical branches of mathematics as analytic geometry, trigonometry, and complex analysis owe their existence to linking metaphors. But before we go on, we should point out an important phenomenon arising from the fact that different linking metaphors can have the same source domain.

Metaphorically Ambiguous Sets Recall for a moment two central metaphors that are vital to the set-theoretical foundations of mathematics: The Ordered Pair metaphor and the Natural Num- bers Are Sets metaphor.

The Ordered Pair Metaphor Source Domain Target Domain Sets Ordered pairs The set {{a}, {a, b}} Æ The ordered pair (a, b)

The Natural Numbers Are Sets Metaphor Source Domain Target Domain Sets Natural numbers The empty set Æ Zero 0465037704-02.qxd 8/23/00 9:51 AM Page 151

Sets and Hypersets 151

The set containing the empty Æ One set {} (i.e., {0}). The set {, {}} (i.e., {0, 1}) Æ Two The set {, {}, {, {}}} Æ Three (i.e., {0, 1, 2}) The set of its predecessors Æ A natural number (built following the above rule)

Within formal mathematics, these metaphors are called “definitions.” The pur- pose of these definitions is to reduce arithmetic to set theory. An “eliminativist” is someone who takes this reduction literally—that is, who believes that these metaphorical definitions eliminate numbers and ordered pairs from the ontology of mathematics, replacing them by sets with these structures. The eliminativist claims that numbers and ordered pairs do not really exist as separate kinds of things from sets; rather, they are sets—sets of the form given in the two metaphors. In the eliminativist interpretation, there is no number zero separate from the empty set. Rather, the number zero in reality is the empty set. And there is no number one separate from the set containing the empty set. Rather, the set containing the empty set is the number one. Similarly, no ordered pairs exist as ordered pairs distinct from sets. Rather, on the eliminativist view, the ordered pair (a, b) really is a set, the set {{a}, {a, {b}}}. Let us consider the following two questions:

1) Exactly what set is the ordered pair of numbers (0, 1)? 2) Exactly what set is the set containing the numbers 1 and 2 as members?

According to the two metaphorical deﬁnitions, the answers should be as follows:

1) The ordered pair (0, 1) does not exist as such. It is the set: {{}, {, {}}}. 2) The set containing the numbers 1 and 2 as members does not exist as such. It is the set: {{}, {, {}}}.

The answer is the same. The ordered pair (0, 1) is a conceptual entity different from the set {1, 2}. One might expect that the reduction of ordered pairs and natural numbers to sets would preserve such conceptual distinctions. It is clear from this example that it does not. Is this a problem for eliminativists? It depends on whether they have such an expectation or not. 0465037704-02.qxd 8/23/00 9:51 AM Page 152

152 Where Mathematics Comes From

From a cognitive point of view, this is no problem. It is common for different concepts to have the same metaphorical source. For example, “You’re warm” can have at least two metaphorical meanings—You are affectionate or You are getting close to the answer. In mathematics, the Ordered Pair metaphor and the Natural Numbers Are Sets metaphor serve the purposes of set theory. Since they preserve inference, inferences about sets will be mapped appropriately onto both numbers and ordered pairs. The moral is simple: Mathematics is not literally reducible to set theory in a way that preserves conceptual differences. However, ingenious metaphors linking ordered pairs to sets and numbers to sets have been explicitly constructed and give rise to interesting mathematics. It is important to distinguish a literal definition from a metaphorical one. Indeed, there are so many conceptual metaphors used in mathematics that it is extremely important to know just what they are and to keep them distinct. Mathematical idea analysis of the sort we have just done clarifies these issues. Where there are metaphorical ambiguities in a mere formalist symboliza- tion like {{}, {, {}}}, it reveals those ambiguities because it makes explicit what is implicit. It also clarifies the concept “same size” for a set, distinguish- ing the everyday notion from that defined by Cantor’s metaphor. And it makes simple and comprehensible the idea of hypersets as an instance of the Sets Are Graphs metaphor.