Central Questions

How can we formalize arithmetic? Specifically, how can we define the set $\mathbb{N}$ of natural numbers? On a more foundational level, is there a need to do so? While $\mathbb{N}$ is arguably a very familiar concept, we will treat them much like we have never seen them before in this article. This know-nothing game and its rule would be very evident by the end of this post.

What is a Natural Number?

Informal Definition

As one can take a glimpse from the word "natural," natural numbers are the ones that are the most natural to humans. (Probably a pun intended.) We count the number of objects using natural numbers, all the coins and bills have certain monetary values signified by a respective natural number, etc. (NB: There is an interesting variation concerning how these numbers are treated in respective languages' number system. I have created some linguistics olympiad problems on precisely this subject matter. For example, refer to the problem on Ngkolumpu, a language that belongs to the Yam family, spoken by approximately 100 people in Papua province, Indonesia.)

The concept of zero and floating point numerals are inventions of following generations.

Why Informal Definition is Insufficient

Assume that we encounter an alien species that have traveled from a distant galaxy several thousand light years away, and further assume that they surprisingly do not have a system of arithmetic. Their way of counting, enumeration, and virtually every activity involving numbers and computations, just work differently. We are endowed with a task to define natural numbers for them as a part of cultural exchange.

It is not difficult to see that "defining" natural numbers with $1$, $2$, $3$, etc. like we did above is insufficient. We can instruct how to read individual numbers and what kind of property each number has, but it is not really a definition per se for a set of natural numbers. Moreover, it is just an enumeration of discrete numbers and cannot be viewed as a formal definition.

Our definition for natural numbers has to apply to every single natural number and introduce seamlessly the central characteristics of natural numbers.

Peano Axioms (PA)

Giuseppe Peano, an Italian mathematician, was one of the first few pioneers who embarked on the journey of defining natural numbers, setting the foundation for mathematical logic. He wished to define what natural numbers are by postulating a set of axioms, which was aptly named the Peano axioms.

There are a few characterizations on Peano axioms as well as variants depending on the type of literature, which subtly differs in the precise number of items in the axiom. In this post, we will treat it in the most general and simplest form without losing too much generality.

PA1: "First" Natural Number

The first axiom states that $1$ belongs to $\mathbb{N}$. This is like a base case in mathematical induction or recursion, yet this has to be made much more precise by the following axioms.

PA2: Successor Function

We define a unary successor function $S$, and $\mathbb{N}$ is closed under $S$. Every succeeding number of an arbitrary element in $\mathbb{N}$ is also an element of $\mathbb{N}$.

Intuitively, one can make a reasonable guess that $S(n) = n+1$, but we cannot say that with certainty in this formal language specification under Peano axiom. This is because we do not have a concept of addition in this system. Further, we do not know any number other than $1$. So, we cannot say that the successor of $1$ is $2$. Instead, the only valid presentation is that $S(1)$ is the successor of $1$.

Since $S(1) \in \mathbb{N}$, $S(S(1)) \in \mathbb{N}$. This pattern continues.

Note that $S(n)$ is also often written as $n^\prime$.

PA3: No Number Succeeds the First Natural Number

The significance of PA3 lies in the fact that, with only PA1 and PA2, we cannot guarantee an infinite number of elements in $\mathbb{N}$. Consider: can we conclude that $1 \neq S(1)$? The response to this inquiry is negative, which provides the rationale that we need some starting point.

Then, with PA3 established, can we say that $1$ is the smallest natural number?

The name for $S$ — the successor function — might be a misnomer: just because of the fact we named it the successor function (which indeed is the case based on our understanding of $\mathbb{N}$), we should not hastily conclude that $1$ is the smallest natural number. This is because we do not have a concept of comparing numbers.

PA4: Equality Relations

Based on the limitation identified above, we need a way of specifying our system further by introducing additional technicalities.

What PA4 purports is that we are defining some kind of operation of determining the successor natural number for an arbitrary natural number. If two natural numbers have the same successor natural number, then these two original natural numbers have to be equal.

PA4 states that the function $S$ is injective.

The equality relation is an equivalence relation; that is, $=$ is reflexive, symmetric, and transitive. In first-order logic,

PA5: Induction

Arguably PA5 is the heart of the Peano axiom. First, we will define an unary predicate $\varphi$, which is some proposition about $\mathbb{N}$.

This is a typical formulation of mathematical induction. Since $\mathbb{N}$ is an infinite set, we bring induction for $\varphi$ to apply for all natural numbers.

Next Step: Peano Arithmetic

Our initial question involved defining arithmetic. Based on PA1 through PA5, we will take a few more steps to complete the discussion on Peano Arithmetic.

Defining Addition

Addition is an operator that takes two natural numbers and maps it to a natural number: $+: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$. With this addition operator, we write out two axioms of addition:

Let’s see how this works for $3 + 2 = 5$ as an example. To not clutter the notations, we will use the prime notation, $n^\prime = S(n)$. First, note that $3 = 1^{\prime\prime\prime}$ and $2 = 1^{\prime\prime}$ and start from it.

Defining Multiplication

Similar to $+$, $\times$ is a binary operator: $\times: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$. Here are the two axioms of multiplication:

This is the typical multiplication operator, which shall not induce any confusions. One can verify that $\textbf{MUL1}$ and $\textbf{MUL2}$ indeed hold by trying out with a few numbers of choice.

Defining Inequality

Lastly, we will set up axioms of inequalities. $<$ is a binary relation on $\mathbb{N}$.

$\textbf{IEQ1}$ states that no natural number is smaller than $1$, while $\textbf{IEQ2}$ specifies two possibilities in relation to the relative magnitude of two natural numbers.

With addition, multiplication, and inequality defined, we can now perform ordinary arithmetic operations on any natural numbers, i.e. the construction for Peano Arithmetic is now complete.