Numbers: Their Tales, Types, and Treasures.

Chapter 11: Numbers and Philosophy

11.5.THE LOGICIST DEFINITION OF A CARDINAL NUMBER

In his book Introduction to Mathematical Philosophy, Bertrand Russell used the concept of a set and the bijection principle (see chapter 1) to define the abstract concept of a cardinal number. This abstract definition makes clear that a number does not refer to any particular group of objects but to the whole class of sets with the same number of objects in it. For this, it is important to remember that in order to determine whether two sets have the same number of elements, it is not necessary to count them; one only has to find a one-to-one correspondence (a bijection) between the elements of the two sets.

One first defines that two sets A and B are equivalent (for the purpose of counting), whenever there is a bijection between these two sets. Thus, when two sets are equivalent, then the elements of one set can be paired with the elements of the other, with no leftover elements in either set. The set of fingers on the right hand is equivalent to the set of fingers on the left hand, the pairing being established by putting the fingertips together. Equivalent sets cannot be distinguished by counting. Hence the definition of “number” must refer to the whole bunch of equivalent sets.

Russell, therefore, defines the “cardinal number of a set A” simply as the collection of all sets that are equivalent to A:

· The class of all sets that are equivalent (for the purpose of counting) is called a cardinal number.

Here we could have said “the set of all sets that are equivalent.” The word “class” is used because in set theory this describes a particular type of (infinite) collection that avoids certain logical problems encountered with arbitrary “sets of sets”—logical problems that were discovered in 1901 by Russell.

Numbers thus become “equivalence classes” of sets. This just means that all sets containing a certain number of elements contribute to the definition of that number. Conceptually, it is the whole collection of equivalent sets that describes best their common property—and this property is the number. According to Bertrand Russell, this is the same process of abstraction that happens in everyday life, where, for example, the best description of what is meant by the abstract concept “table” is the whole collection of all objects that are called “table.” Only in that way could the abstract notion “table” encompass everything that might be called “table.”

Russell describes his idea of reducing the concept of number to set theory with the following words: “We naturally think that the class of couples (for example) is something different from the number 2. But there is no doubt about the class of couples: it is indubitable and not difficult to define, whereas the number 2, in any other sense, is a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down. It is therefore more prudent to content ourselves with the class of couples, which we are sure of, than to hunt for a problematical number 2 which must always remain elusive.”¹²

Thus, according to Russell, the number 2 is the collection of all pairs—it consists of all sets that contain precisely two elements because all these sets are indistinguishable by counting and are, therefore, considered equivalent. So, we consider the whole collection of sets that are equivalent to a pair of shoes and call it “number 2.” Any particular set of two objects is then just an example of the number 2, a representative, very much in the same sense as any particular dining table is just a representative of the abstract notion “table.”

At first sight, this definition might seem circular, because how could we define the collection of all two-element sets unless we already know what “two” is. But actually, one can define a two-element set without mentioning the number 2, as follows: We say that a set A contains two elements, whenever the following conditions are fulfilled:

(a)A contains an element x and an element y, such that x is not equal to y,

(b)for all elements z belonging to A we have either z = x or z = y.

These conditions express in purely logical terms (as an equality), what we mean by a set of two elements. In an analogous way, one can define a set of three elements, four elements, and so on. Thus, the goal seems to have been reached to define the finite numbers on the basis of pure logic.

Hungarian-American mathematician John von Neumann (1903–1957) described a purely set-theoretic method to construct the natural numbers. In the axioms of mathematical set theory, there is a unique set that contains no elements at all. It is called the “empty set” or “null set” and denoted by Ø or occasionally by { }. Neumann took the empty set Ø to represent the number 0. Next, we can form the set that contains the empty set as its only element; this is the set {Ø}. We take it to represent the number 1 because it obviously contains precisely one element. Next, we can form the set with the elements Ø and {Ø}, that is {Ø,{Ø}}. Remember that the collection of all sets equivalent to this set would be number 2. For the number 3, we combine all previously obtained objects, namely Ø, {Ø}, and {Ø,{Ø}}, into a new set and proceed in an analogous way. In this way, we construct a sequence of certain prototypical sets using only the basic notions of set theory, nothing else, and the numbers are then the collections of all sets that are equivalent to the prototypical sets.

0…represented by…Ø (empty set)
1…represented by…{Ø}
2…represented by…{Ø, {Ø}}
3…represented by…{Ø, {Ø}, {Ø, {Ø}}}
4…represented by…{Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø, {Ø}}}}

and so on.

If x is a set constructed in this way, then the successor of x is always defined as the union of x and {x}. On the nth stage, x is a set containing n elements, and the set {x} contains just one element, namely x. The union of these two sets will, thus, contain n + 1 elements. This will serve as a new prototypical set with n + 1 elements representing the number n + 1. All other sets that are in one-to-one correspondence with this prototypical set would together be the “number n + 1.” In that way, natural numbers are created, one after another, out of the empty set Ø, that is, “out of nothing.” We also note that the cardinal numbers thus obtained are naturally ordered by size.

From here, it is still a long way to a complete mathematical exposition of numbers and their arithmetic, or even to a precise mathematical definition of the set of all natural numbers. This approach actually needs a lot of experience in abstract logical reasoning, and we shall not pursue it any further here. We should note that in most situations not even a mathematician would think of the number 4 as the “class of all sets equivalent with {Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø, {Ø}}}}.” This construction, nevertheless, serves to show that the abstract notion of a number can be defined on a strictly logical basis, using only very elementary definitions from set theory. The mathematical definition alluded to here formalizes the somewhat vague statement, “The number 4 describes what 4 apples and 4 people have in common.”

The concepts described here have paved the way to a logically rigorous analysis of mathematically advanced ideas about infinity, leading to definitions of infinite cardinal numbers and infinite ordinal numbers that have opened up a huge field of research for the mathematicians of the twentieth century.