8.1: The Natural Numbers
What are the real numbers and why don’t the rational numbers suffice for our mathematical needs? Ultimately the real numbers must satisfy certain axiomatic properties which we find desirable for interpreting the natural world while satisfying the mathematician’s desire for a formal foundation for mathematical reasoning.
Of course the real numbers must contain the rational numbers. We also require that the real numbers satisfy rather obvious algebraic properties which hold for the rational numbers, such as commutativity of addition or the distributive property. These axioms allow us to use algebra to solve problems. Additionally we must satisfy geometric properties like the triangle inequality which permit the interpretation of positive real numbers as distances. We need our number system to contain numbers that arise from the algebraic and geometrical interpretation of numbers. Unfortunately the rational numbers do not suffice for this limited objective. For instance, \(\sqrt{2}\) , which you know by Example \(3.23\) to be irrational, arises geometrically as the length of the diagonal of the unit square, and as the solution to the algebraic equation \(x^{2}=2\) .
The development of the limit gave rise to new questions about the real numbers. In particular, when are we assured that a sequence of numbers is convergent in our number system? The proof of convergence claims often use another property of the real numbers, the least upper bound property. Many of the powerful conclusions of calculus are consequences of this property. Loosely speaking, the least upper bound property implies that the real number line doesn’t have any "holes". Put another way, if all the elements of one non-empty set of real numbers are less than all elements of another non-empty set of real numbers, then there is a real number greater than or equal to all the elements of the first set, and less than or equal to all the elements of the second set. This property is called order-completeness, and is formally defined in Section 8.10. Order-completeness, and its desirable consequences, do not hold for the rational numbers.
How do we prove the existence of a set with order and operations that satisfies all these needs simultaneously? One cannot simply assume that such a structure exists. It is possible that the properties specified are logically inconsistent. We might attempt to construct the set. What are the rules for the construction of a mathematical object? This question prompted mathematicians of the late nineteenth and early twentieth century to develop the rules for such a construction - the axioms of set theory.
For this reason we build the real numbers with a set-theoretic construction. That is, we shall construct the natural numbers, integers, rational numbers and irrational numbers in turn, using basic sets, functions and relations. In so doing we shall construct a set with order and operations that contains the rational numbers (or a structure that behaves precisely like we expect the rational numbers to behave), satisfies the algebraic and order axioms, has the properties we need for calculus and is constructed with the tools that you developed in Chapters 1 and 2 .
The Natural Numbers
When we introduced the natural numbers in Chapter 1 we were explicit that we were not defining the set. Instead we proceeded under the assumption that you are familiar with the natural numbers by virtue of your previous mathematical experience. Now we define the natural numbers in the universe of sets, constructing them out of the empty set. DEFINITION. Successor function Let
\(Y\)
be a set. The successor function,
\(S\)
, is defined by
\[S(Y):=Y \cup\{Y\} .\]
DEFINITION. Inductive set Let
\(S\)
be the successor function and
\(X\)
be any collection of sets satisfying the following conditions:
(1)
\(\emptyset \in X\)
(2)
\([Y \in X] \Rightarrow[S(Y) \in X]\)
.
Then \(X\) is called an inductive set.
DEFINITION. Natural numbers Let \(X\) be any inductive set. The set of natural numbers is the intersection of all subsets of \(X\) that are inductive sets.
Are the natural numbers well-defined? That is, does the definition depend on the choice of the set \(X\) ? If \(\mathcal{F}\) is a family of sets, all of which are inductive, it is easily proved that the intersection over \(\mathcal{F}\) is also inductive. If we are given sets \(X\) and \(Y\) that are inductive, will the sets give rise to the same set of "natural numbers"? Again it is easily seen that the answer is yes since \(X \cap Y\) is a subset of both \(X\) and \(Y\) , and is inductive. The "natural numbers" defined in terms of \(X\) and \(Y\) will be the "natural numbers" defined in terms of \(X \cap Y\) - they constitute the "smallest" inductive set. In order to define the natural numbers in the universe of sets, it must be granted that there exists an inductive set. It is an axiom of set theory that there is such a set, called the axiom of infinity (see Appendix B for a discussion of the axioms of set theory).
What does this set have to do with the natural numbers as we understand and use them in mathematics? Consider the function, \(i\) , defined by \[i(0)=\emptyset\] and \[i(n+1)=i(n) \cup\{i(n)\} .\] So \[\begin{aligned} i(0) &=\emptyset \\ i(1) &=\{\emptyset\} \\ i(2) &=\{\emptyset,\{\emptyset\}\} \\ i(3) &=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\} . \end{aligned}\] Then \(i\) gives a bijection between the natural numbers, as we understand them intuitively, and the minimal inductive set which we defined above.
Let us define \(\ulcorner n\urcorner\) formally as the set one obtains by applying the successor function \(S\) to the empty set \(n\) times. So \[0=\emptyset\] and for \(n>0\) the set \[\ulcorner n\urcorner=\{\emptyset,\{\emptyset\}, \ldots\}\] has exactly \(n\) elements, and we shall identify it with the set \[\{0,1, \ldots, n-1\}\] that we earlier chose as the canonical set with \(n\) elements.
The set \[\mathbf{N}:=\{\ulcorner n\urcorner \mid n \in \mathbb{N}\}\] is inductive, and therefore contains the natural numbers. Finally the reader may confirm that \(\mathbf{N}\) has no proper subset that is inductive.
To summarize the construction so far, the axiom of infinity guarantees that there is a set that is inductive. Pick such a set, \(X\) . The intersection of all subsets of \(X\) that are inductive is \(\mathbf{N}\) , which we can identify with the natural numbers (conceived intuitively) by the bijection i. In order to continue the construction, we consider \(\mathbb{N}\) and \(\mathbf{N}\) to be the same set. We need \(\mathbb{N}\) to have the operations \(+\) and . as well as the relation \(\leq\) . We shall define addition in \(\mathbb{N}\) with basic set operations and cardinality. If \(m, n \in \mathbb{N}\) , then we define addition by \[m+n:=|(\ulcorner m\urcorner \times\{\ulcorner 0\urcorner\}) \cup(\ulcorner n\urcorner \times\{\ulcorner 1\urcorner\})| .\] Recall that the cardinality of a finite set is the unique natural number that is bijective with the set - hence the complicated expression on the right hand side of the definition is a natural number. It is easy to confirm that addition defined in this manner agrees with the usual operation in \(\mathbb{N}\) . Why would we bother to define an operation you have understood for many years? We have defined addition of natural numbers as a set operation.
Multiplication is somewhat easier to define. If \(m, n \in \mathbb{N}\) , then \[m \cdot n:=|\ulcorner m\urcorner \times\ulcorner n\urcorner| .\] (Of course, by \(\ulcorner m\urcorner \times\ulcorner n\urcorner\) we mean the Cartesian product of the sets \(\ulcorner m\urcorner\) and \(\ulcorner n\urcorner\) . ) Finally if \(m, n \in \mathbb{N}\) \[[m \leq n] \Longleftrightarrow[\ulcorner m\urcorner \subseteq\ulcorner n\urcorner] .\] You should confirm that the operations and the relation agree with the usual \(+, \cdot\) and \(\leq\) on the natural numbers.
Having completed this construction it is reasonable to ask whether \(\mathbb{N}\) is truly the set of natural numbers. It is certainly justifiable for you to conclude that no clarity about the number 2 is provided by identifying it with the set \(\{\emptyset,\{\emptyset\}\}\) . What we gain is a reduction of numbers to sets that will carry us through the construction of all real numbers, including numbers that are not easy to construct.