authored by Premmi and Beguène

Previous Topic: An Axiomatic Study of Numbers

Introduction

Thinking of numbers intuitively brings to mind the simplest and most fundamental set of numbers, namely the set of natural numbers. These numbers are used to count objects like cars, books, pens, etc. If we associate natural numbers such as 1, 2, 3, etc. with counting, then with what corresponding concepts do we relate numbers like -4, \sqrt{3} \text{ and } \frac{22}{7}?

To reason about all kinds of numbers encountered during our study of mathematics, we need a precise mathematical framework for defining numbers. We will build this framework by first rigorously defining natural numbers axiomatically without relying on the intuitive notion of counting. Then, using this framework as a foundation, we will construct all the other sets of numbers such as integers, rational numbers, real numbers and complex numbers in terms of the natural numbers.

A good axiomatic system assumes as little as possible, while proving as much as possible. To create an efficient axiomatization for the natural numbers, we must distill these numbers to their essential properties. Intuitively, we understand various aspects of the natural numbers, such as their existence and the basic properties of the binary operations addition, multiplication, and the ‘less than’ relation. How few of these concepts can we take as axioms, from which we can deduce everything else that we need to know about the natural numbers? It turns out that remarkably little is required for the axiomatization of the natural numbers—neither addition, nor multiplication, nor the ‘less than’ relation need to be taken as axioms; these will all be constructed from our fundamental axioms.

The standard axiomatization of the natural numbers, know as the Peano Axioms, was originally formulated by the Italian mathematician, Giuseppe Peano. In 1889, Peano developed the real number system based on his axioms for the natural numbers. He defined natural numbers through nine axioms, four of which established the properties of the equality relation “=” with regard to natural numbers, while the remaining five axioms provided a complete and rigorous definition of natural numbers.

In order to appreciate the intellectual feat of Peano, it’s worth noting that utilizing only his axioms, we are able to prove all the established properties of the natural numbers. Furthermore, these axioms facilitate the construction of the integer, rational, real, and complex number systems.

Before we discuss Peano’s Axioms in detail, it is a useful exercise to explore an alternative way to describe natural numbers, distinct from the usual intuitive notion of counting. Such an exploration would help us to independently arrive at Peano’s axioms.

Intuition behind the Axiomatic Definition of Natural Numbers

How do we model the concept of natural numbers denoted by 0, 1, 2, 3 and so on without relying on the notion of counting?

One way would be to take a set-theoretic approach. That is, we can define a set of natural numbers by systematically enumerating the properties and relationships of the members of this set such that the set results in \{0, 1, 2, 3, \ldots\}.

Let us denote the set of natural numbers as \mathcal{N}. First, we recognize that 0 should be a part of this set. Next, we want 0 to lead us to 1, 1 to 2 and we should be able to continue this way, naming each successive number as far as we wish. This is illustrated below.

0 \rightarrow1 \rightarrow 2 \rightarrow 3 \rightarrow \ldots

From the above diagram we can see that to model this relationship we need a “next” operation that given a natural number, produces the next natural number in the sequence.

The above diagram can also be viewed as shown below.

\begin{equation*} 
\begin{split}
0 &\rightarrow 1 \\
1 &\rightarrow 2 \\
2 &\rightarrow 3 \\
\vdots &\quad\,\,\,\, \vdots \\
\end{split}
\end{equation*} 

We can see from the diagram that an input of 0 yields an output of 1, an input of 1 yields 2 and so on. Therefore, we can model the “next” operation as a function S : \mathcal{N} \rightarrow \mathcal{N} that takes a natural number as input and produces a natural number as output. Here, the letter S stands for ‘successor’ and we have S(0) = 1, S(1) = 2 and so forth. We will refer to this function S as the successor function since it establishes a succession within the set of natural numbers, \mathcal{N}. This function S is illustrated below.

From the diagram above, we can observe that the successor function S has the following properties:

1. Not Surjective: There is no natural number in \mathcal{N} that, when given as input to the successor function S, results in 0 as output. This implies that not every element of the codomain of S is the image of at least one element from its domain. Therefore, we can conclude that S is not surjective because S(n) \neq 0 \text{ for any } n \in \mathcal{N}.
2. Injective: Different inputs to S yield different outputs. This means that every element of the codomain of S is the image of at most one element from its domain; that is, S(m) = S(n) \text{ implies that } m = n \text{ for any } n, m \in \mathcal{N}. Hence, S is injective.
   

From the diagram, we can also see that a natural number is either 0 or can be obtained from 0 by applying the successor function to 0 a finite number of times. This implies that \mathcal{N} is the minimal non-empty set that contains 0 and admits a successor function satisfying conditions (1) \text{ and } (2).

Therefore, for the Peano Axioms to accurately describe the set of natural numbers, they must define a set that contains 0 and admits a successor function as described above.

Original Formulation of Peano’s Axioms

Before we discuss the modern version of Peano’s Axioms, it is interesting to know how these axioms were originally stated by Giuseppe Peano. While perusing Peano’s Axioms it is worth keeping in mind that during Peano’s time, the concept of set was still nascent; it was Peano who introduced the symbol \in in 1889 to denote “is an element of”. Being aware of Peano’s original formulation of these axioms helps us appreciate how far we have come in our mathematical journey and highlights that our journey towards better mathematical notation and abstraction still continues. 😃

The axioms stated below appear between the pages 387-408 in the book “Historia Mathematica \mathit{1}” published in the year 1974.

In 1891, two years after the publication of his axioms for the natural numbers, Giuseppe Peano published an article titled “Sul concetto di numero” which translates to “On the Concept of Number,” in a journal he founded that same year. In this article, he reduced his list of axioms to five by eliminating the four axioms related to the equality relation “=\!\!".

The following are the five Peano’s axioms stated in the article :

We can see how the archaic notation obfuscates these axioms.

These maybe interpreted as :

In the Peano Axioms published in 1889 \text{ and } 1891, the sequence of natural numbers began with 1, and the set of natural numbers was denoted by \mathit{N}. However, in 1898 these axioms were modified so that the sequence began with 0 and the set was denoted by \mathit{N_{\!0}}.

The set of five Peano Axioms was increased to six in 1901 with the addition of the axiom – \mathit{N_{\!0}} \in \text{Cls}, i.e., the natural numbers form a class. With the addition of this last axiom, the axioms have received their final form, which are listed below.

The Axiomatization of Natural Numbers

We will first define the notion of equality as it pertains to natural numbers, and then we will formulate the Peano Axioms such that it provides an axiomatic definition of natural numbers.

Although Giuseppe Peano omitted the four axioms related to the equality relation from his later formulation of the axioms regarding natural numbers, we will still discuss these axioms for the sake of completeness. The likely reason for this omission is because the concept of equality extends beyond natural numbers to mathematical objects in general and pertains more to the realm of logic, which specifies the conditions under which two mathematical objects can be considered equal.

In our discussion of Peano’s Axioms, we will adopt some conventions from its final form. Therefore, we will start with 0 instead of 1 in our axiomatic system and denote the set of natural numbers by \mathbb{N_0}, where the subscript 0 reminds us that 0 is included.

The Notion of Equality

Before we define the set of natural numbers \mathbb{N_0} axiomatically, we will formalize the notion of equality through the four axioms of Peano, which establish the properties that the equality relation, denoted by =, must satisfy.

Suppose there exists a set \mathbb{N_0} that satisfies Axioms 1 t0 9 listed below.

Firstly, every natural number should be equal to itself; this is called the reflexivity axiom.

Axiom \mathbf{1}. For every x \in \mathbb{N_0}, x = x.

Secondly, if one natural number equals a second one, then the second one should equal the first one. This is known as the symmetry axiom.

Axiom \mathbf{2}. For every x, y \in \mathbb{N_0}, \text{ if } x = y, \text{ then } y = x.

Thirdly, if one natural number is equal to a second, and that second natural number is equal to a third, then the first and third are equal to each other. This is called the transitivity axiom.

Axiom \mathbf{3}. For every x, y, z \in \mathbb{N_0}, \text{ if } x = y \text{ and } y = z, \text{ then } x = z.

These three properties of reflexivity, symmetry and transitivity are applicable to any two mathematical objects that are related by the equality. As we have already discussed during our study of Set Theory, equality is an example of an equivalence relation, which is a type of homogeneous binary relation that satisfies the above three properties of reflexivity, symmetry and transitivity.

Since the equality relation is defined generically for all mathematical objects and not just for natural numbers, we must make explicit the assumption that if two mathematical objects are equal—i.e., they satisfy the equality relation—and one of them is a natural number, then the other must also be a natural number. The next axiom makes this assumption explicit, namely, a natural number can only be equal to another natural number.

Fourthly, if any mathematical object is equal to a natural number, then that mathematical object is itself a natural number. This is called the closure of equality axiom.

Axiom \mathbf{4}. For all x \text{ and } y, \text{ if } x \in \mathbb{N_0} \text{ and } x = y, \text{ then } y \in \mathbb{N_0}.

That is, the set of natural numbers is closed under equality.

The Peano Axioms

We will now discuss the five main Peano axioms that define the natural numbers. Peano aimed to formulate these axioms such that the fewest possible axioms could generate all the natural numbers. Therefore, we will construct the Peano axioms by checking, after stating each axiom, whether the axioms stated thus far can unambiguously result in the set of natural numbers that we know of. That is, we will continue constructing the Peano axioms until these axioms, when taken together, incontrovertibly result in \mathbb{N}_0 = \{0, 1, 2, \ldots\}.

This method of constructing the Peano axioms leads to the insight that the entire set of natural numbers can be generated by asserting the existence of at least one natural number and then defining a function called the successor function. This function takes a natural number as input and outputs another natural number, resulting in the construction of all remaining natural numbers.

Let us now proceed with the construction of the Peano Axioms.

Since we start counting from 0, it is unsurprising that 0 is the most obvious element to axiomatically include in the set of natural numbers.

Fifthly, 0 is a natural number.

Axiom \mathbf{5}. 0 \in \mathbb{N_0}.

Thus far we are only guaranteed the existence of a single natural number, 0.

From 0 we should be able to generate the other natural numbers; that is starting with 0 we should be able to reach 1, from 1 reach 2 and so on, akin to counting. We can model this progression from one natural number to the next using a function that takes a natural number as input and produces another natural number as output. This function is called a successor function since it establishes a succession within the set of natural numbers and is written as S : \mathbb{N_0} \rightarrow \mathbb{N_0}. The next axiom simply states that there is a function S whose domain and codomain are the set of natural numbers, \mathbb{N}_0.

Sixthly, every natural number has a successor which is also a natural number.

Axiom \mathbf{6}. If x \in \mathbb{N}_0, then S(x) \in \mathbb{N}_0.

That is, the set of natural numbers, \mathbb{N}_0, is closed under the successor operation, S.

As this axiom implies, we will refer to S(x) as the successor of \mathit{x}.

Till now we have only established that the set of natural numbers contains 0 and its successor, S(0), where the function S takes natural numbers as input and outputs natural numbers. However, we are still quite far from having the set of natural numbers that we know of, since we could have \mathbb{N}_0 = \{0\} and define S(0) = 0, which would still satisfy all of the above axioms. In this case, \mathbb{N}_0 = \{0\}, but we want \mathbb{N}_0 = \{0, 1, 2, 3, \ldots\}.

To achieve this, we need to ensure that the successor function S does not output 0. Our next axiom will guarantee this by forbidding 0 from being the successor of any natural number, including itself.

Seventhly, 0 is not the successor of any natural number.

Axiom \mathbf{7}. For every natural number x \in \mathbb{N}_0, S(x) \neq 0.

That means that there is no natural number whose successor is 0. Consequently, the preimage of 0 under S defined on the set of natural numbers is an empty set.

As a consequence of this axiom, we know that S(0) \neq 0. Therefore, S(0) must equal some other natural number, which we can denote by 1. Hence, we can define 1 by S(0) = 1.

Based on axioms 5, 6 \text{ and } 7 we are guaranteed the existence of at least two natural numbers, 0 \text{ and } 1, but not necessarily others.

For example, we could define \mathbb{N}_0 = \{0, 1\}, where S(0) = 1 \text{ and } S(1) = 1. In this case, both natural numbers 0 \text{ and } 1 have the same successor, which is 1.

If S(0) = 1 \text{ and } S(1) = 1, then the two natural numbers 0 \text{ and } 1 have the same successor, 1. This set of natural numbers together with the successor function S defined on it would still satisfy all the above axioms. However, if we stop here, the axioms constructed thus far do not guarantee the existence of the rest of the natural numbers that we know of, namely, 2, 3, 4 \ldots.

Therefore, our next axiom should ensure that different natural numbers have different successors. This means that every natural number is the successor of at most one natural number (since 0 is not the successor of any natural number) which implies that S must be an injective function.

Eighthly, no two natural numbers have the same successor unless they are equal.

Axiom \mathbf{8}. For all x, y \in \mathbb{N}_0, if S(x) = S(y), then x = y.

This axiom leads to some important consequences. It excludes the possibility of defining \mathbb{N}_0 to be just \{0, 1\}. We already have S(0) = 1 and since S is an injective function, we cannot have S(1) = 1. Axiom 7 excludes the possibility that S(1) = 0. Thus S(1) must be some other natural number, which we denote as 2. Therefore, we can define 2 = S(1).

By a similar argument, S(2) cannot be 0, 1 \text{ or } 2. Hence, it must be some other natural number, which we denote as 3. Continuing this way, we see that \mathbb{N}_0 must contain all the natural numbers that we know of.

So far we have established that \mathbb{N}_0 must include 0, its successor 1 = S(0), its successor’s successor 2 = S(1) and so on. Thus \mathbb{N}_0 must include 0, S(0), S(S(0)), S(S(S(0))), \ldots. In order to avoid so many nested applications of S we use the numerals 1, 2, 3 to denote S(0), S(S(0)) \text{ and } S(S(S(0))), respectively.

These first eight axioms have resulted in the definition of \mathbb{N}_0 to include all the natural numbers that we know of.

Therefore, so far we only know that

\{0, 1, 2, \ldots\} \subset \mathbb{N}_0

At this point, it is interesting to ask whether our axiomatic definition of \mathbb{N}_0 precludes the inclusion of additional elements.

In order to answer this question, let us consider a version of \mathbb{N}_0 that satisfies all the above axioms but is not the usual set of natural numbers that we know of. That is,

\mathbb{N}_0 = \{0, 1, 2, 3, \ldots\} \cup \{!, -\}

As can be seen, this version of \mathbb{N}_0 contains all the natural numbers and also includes two other symbols, ! \text{ and } -.

We will next define the successor function defined on this set. For the subset \{0, 1, 2, 3, \ldots\} of \mathbb{N}_0, we define S as we have described above i.e., S(0) = 1, S(1) = 2, S(2) = 3 and so on. For the subset \{!, -\} of \mathbb{N}_0, we define S(!) = - \text{ and } S(-) = !.

This version of \mathbb{N}_0 with this successor function satisfies all the axioms, but it has more elements than what we want our set of natural numbers to have. This is shown below.

Similarly, there could be other versions of \mathbb{N}_0 with different successor functions, each of which satisfies all of the above axioms from 5 \text{ through } 8 but could also have elements other than natural numbers. This is illustrated below.

Based on axioms 5 \text{ to } 8, the set \mathbb{N}_0 of natural numbers satisfies the following conditions:

• 0 \in \mathbb{N}_0; and
• if x \in \mathbb{N}_0,\text{then } S(x) \in \mathbb{N}_0, where S(x) denotes the successor of x.

This way of defining a set, where a base clause specifies the basic element of the set and an inductive clause details how to generate additional elements, is called an inductive definition of the set, and such a set is referred to as an inductive set.

Therefore, the axioms 5 \text{ to } 8 only ensure that \{0, 1, 2, \ldots\} \subset \mathbb{N}_0, where \mathbb{N}_0 is any set such that 0 \in \mathbb{N}_0 and if x \in \mathbb{N}_0,\text{then } S(x) \in \mathbb{N}_0.

However, as discussed earlier and shown in the diagram above, this definition of \mathbb{N}_0 does not exclude elements other than natural numbers from being contained in the set.

To ensure that only natural numbers are included in the set \mathbb{N}_0, i.e., \mathbb{N}_0 = \{0, 1, 2, \ldots\}, our next (and final) axiom should declare \mathbb{N}_0 to be the minimal set that satisfies axioms 5 \text{ through } 8; specifically, \mathbb{N}_0 is the intersection of all sets that satisfy these axioms.

Since axiom 9 uses the inductive definition of a set in its construction, it is referred to as the Axiom of Induction.

We will now state our ninth and final axiom. 😇

Axiom \mathbf{9} (Axiom of Induction). If T \subset \mathbb{N}_0 is such that:

• 0 \in T; and
• x \in T \implies S(x) \in T for all x \in \mathbb{N}_0,

then T = \mathbb{N}_0.

As we have already discussed, the axioms 5 \text{ through } 8 ensure that \{0, 1, 2, \dots\} \subset \mathbb{N}_0. Suppose T = \{0, 1, 2, \dots\}. We can see that 0 \in T \text{ and } x \in T \implies S(x) \in T \text{ for all } x \in \mathbb{N}_0. Therefore, by Axiom 9, \{0, 1, 2, \dots\} = \mathbb{N}_0.

Thus we finally have the set of natural numbers that we know of, namely, \mathbb{N}_0 = \{0, 1, 2, \dots\}.

Alternate Formulations of Axiom of Induction: Set-Based and Predicate-Based Perspectives

For the sake of completeness, we will discuss two alternate ways to formulate Peano’s Ninth Axiom, namely the Axiom of Induction.

The Axiom of Induction is a cornerstone of number theory, providing a powerful tool for proving statements about natural numbers. It can be expressed in two equivalent forms: set-based and predicate-based. Each form offers a different perspective on the same fundamental principle.

Set-Based Axiom of Induction: Focusing on Subsets

Axiom \mathbf{9} (Axiom of Induction). If T \subset \mathbb{N}_0 is a set such that:

• 0 \in T; and
• x \in T \implies S(x) \in T for all x \in \mathbb{N}_0,

then \mathbb{N}_0 \subset T.

In simpler terms, if a subset of natural numbers contains 0 and is closed under the successor operation (meaning that if a natural number is in the subset, its successor is also in the subset), then that subset must contain all natural numbers.

As discussed above, Axioms 5 \text{ through } 8 ensure that \{0, 1, 2, \ldots\} \subset \mathbb{N}_0. Additionally, we observe that the set \{0, 1, 2, \ldots\} satisfies the following two conditions of Axiom 9:

• It contains 0; and
• Whenever it contains an element x, it also contains its successor, namely, S(x).

Therefore, by Axiom 9, it follows that \mathbb{N}_0 \subset \{0, 1, 2, \ldots\}.

Since Axioms 5 \text{ through } 8 ensure that \{0, 1, 2, \ldots\} \subset \mathbb{N}_0 and by Axiom 9 we have show that \mathbb{N}_0 \subset \{0, 1, 2, \ldots\}, it follows that \mathbb{N}_0 = \{0, 1, 2, \ldots\}.

This perspective emphasizes the structure of the subset itself. We are concerned with the elements that are members of the subset and how they relate to each other through the successor function.

Predicate-Based Axiom of Induction: Focusing on Properties

Peano’s axioms five through eight collectively define a superset of natural numbers, specifically \{0, 1, 2, \ldots\} \subset \mathbb{N}_0. To ensure that this set \mathbb{N}_0 includes only natural numbers, Peano’s ninth axiom can also be formulated as the principle of mathematical induction over natural numbers. This formulation is equivalent to the axiom of induction and serves the same purpose of removing unwanted elements from the superset \mathbb{N}_0, ensuring that it contains only natural numbers.

The predicate-based form of the Axiom of Induction shifts the focus from subsets to properties (expressed as predicates) that natural numbers may or may not possess.

The reformulation of Axiom 9 in terms of predicates results in the Principle of Mathematical Induction which is stated as follows.

Axiom \mathbf{9} (Principle of Mathematical Induction). For any predicate P(n), where n is a natural number, if:

• P(0) is true, and
• for every natural number n, P(n) being true implies that P(S(n)) is also true,

then P(n) is true for every natural number n.

Here, we are concerned with a property (represented by the predicate P(n)) that natural numbers might possess. If 0 has the property, and if natural number n having the property implies that its successor S(n) also has the property, then all natural numbers must have the property.

This perspective emphasizes the properties of individual natural numbers. We are concerned with whether a given natural number has a specific property.

Equivalence and Connection

The set-based and predicate-based forms are logically equivalent, meaning they express the same fundamental principle. This equivalence is rooted in the relationship between subsets and predicates, as established by the Axiom of Separation.

• From Subset to Predicate:
  • Given T \subset \mathbb{N}_0, it implies from the Axiom of Separation that there exists a predicate P(n) such that P(n) is true if and only if n \in T.
    
• From Predicate to Subset:
  • Given a predicate P(n) and a set \mathbb{N}_0, we can define a subset T = \{n \in \mathbb{N}_0 \,|\, P(n) \text{ is true}\}, by the Axiom of Separation.

Thus, the subset and predicate perspectives are simply two ways of expressing the same fundamental idea. The set-based form emphasizes the elements within a collection, while the predicate-based form emphasizes the properties of individual elements. The Axiom of Separation is the bridge that allows us to move seamlessly between these perspectives.

Why Both Forms Are Useful

Both forms of the Axiom of Induction are valuable tools in mathematical proofs. The set-based form is often used in set theory and related areas, while the predicate-based form is commonly used in number theory and other branches of mathematics where properties of numbers are the main focus.

Essentially, they are two sides of the same coin, and which one to use depends on the context of the problem and the preference of the mathematician.

Method of Definition of Natural Numbers using Peano’s Axioms

It should be noted that Peano’s Axioms only describe how to construct the set of natural numbers and do not define what natural numbers are intrinsically. A particular natural number is given when its generation under the inductive definition is given.

For example, the natural number 2 is defined as that mathematical object which is obtained by starting with the initial object 0 and applying the successor function once and then again i.e., 2 represents S(S(0)).

Existence of the Set of Natural Numbers satisfying Peano’s Axioms

How do we establish the existence of a set, an element of that set, and a function from the set to itself, that satisfy Peano’s Axioms? These axioms themselves are insufficient to prove this existence. Consequently, there are two approaches to resolving this matter.

One common approach in mathematics is to take something as axiomatic and then use it as the basis upon which we prove all our other results. Hence, such an approach requires us to be satisfied with taking the existence of a set satisfying Peano’s Axioms axiomatically. This axiom is called the existence axiom for natural numbers and guarantees that there exists a set with the properties that Peano’s axioms ascribe to it.

The statement of this axiom is as follows:

Existence Axiom for Natural Numbers: There exists a set \mathbb{N}_0 satisfying Axioms 1 \text{ through } 9.

Alternatively, if we use the Zermelo-Fraenkel Axioms as our foundation for set theory, we can prove that something satisfying Peano’s Axioms exists, so we don’t need to assume it separately. We’ll show how this is done later.

Proving Properties of Natural Numbers

Having established the existence of the natural numbers and their fundamental properties as defined by Peano’s Axioms, we will next explore how to rigorously prove that the natural numbers satisfy any specific property using the Axiom of Induction.

Next Topic: Proving Properties of Natural Numbers Using Proof by Induction