0465037704-02.qxd 8/23/00 9:51 AM Page 140 7 Sets and Hypersets 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 no- tion 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 men- tal 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 conceptualiza- tion. 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, con- sider 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 or- dered 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 metaphor- ically define relations, functions, and so on in terms of sets. One of the most in- teresting 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 con- taining 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 conceptu- alize 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 (arith- metic) 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 mem- ber 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 arith- metic. 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 con- ceptual 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.