(or a credit card) when we purchase a muffin for $2 and a frappuccino for
$3. Nonetheless, “three” is like pornography—we know it when we see it,
but damned if we can come up with a great definition of it.More, Less, and the Same
How do you teach a child what a tree is? You certainly wouldn’t start with
a biologist’s definition of a tree—you’d simply take the child out to a park
or a forest and start pointing out a bunch of trees (city dwellers can use
books or computers for pictures of trees). Similarly with “three”—you
show the child examples of threes, such as three cookies and three
stars. In talking about trees, you would undoubtedly point out common
aspects—trunks, branches, and leaves. When talking about threes to
children, we make them do one-to-one matching. On one side of the page
are three cookies; on the other side, three stars. The child draws lines
connecting each cookie to a different star; after each cookie has been
matched to different stars, there are no unmatched stars, so there are the
same number of cookies as stars. If there were more stars than cookies,
there would be unmatched stars. If there were fewer stars than cookies,
you’d run out of stars before you matched up all the cookies.
One-to-one matching also reveals a very important property of finite
sets: no finite set can be matched one-to-one with a proper subset of itself
(a proper subset consists of some, but not all, of the things in the original
set). If you have seventeen cookies, you cannot match them one-to-one
with any lesser number of cookies.
The Set of Positive Integers
The positive integers 1, 2, 3,... are the foundation of counting and arith-
metic. Many children find counting an entertaining process in itself, and
sooner or later stumble upon the following question: Is there a largest
number? They can generally answer this for themselves—if there were a
largest number of cookies, their mother could always bake another one. So
there is no number (positive integer) that describes how many numbers
(positive integers) there are. However, is it possible to come up with some-
thing that we can use to describe how many positive integers there are?
There is—it’s one of the great discoveries of nineteenth-century mathe-
matics, and is called the cardinal number of a set. When that set is finite,
it’s just the usual thing—the number of items in the set. The cardinal
number of a finite set has two important properties, which we discussed
in the last section. First, any two sets with the same finite cardinalThe Mea sure of All Things 15