This post is part of the What is Maths series.

In the late 19th century Peano proposed a set of axioms for Number Theory, which is the study of non-negative whole numbers (i.e. 0, 1, 2, 3, …). His axiomatisation of this field of maths was not only one of the earliest efforts at axiomatisation, but was also so successful that is still used today with only minor modifications. As a side note, the earliest known axiomatisation effort was Euclid's geometry, over 2000 years earlier. I still, however, feel justified in saying that Peano’s is one of the earliest because there weren’t really any attempts between Euclid and the 19th century. It would be fair to say that Euclid was well ahead of his time.

Peano claims that there is a collection of ‘things’ we can refer to as natural numbers, and an operation, called \( S \), that can be applied to these numbers. If \( n \) is a natural number, then we would write \( S ( n ) \) to refer to the result of applying \( S \) to \( n \). Formally, we call \( S \) the ‘successor’ operation, but informally we can think of it as adding 1. I will typically omit the parentheses, writing \( S n \) instead of \( S ( n ) \), for the sake of clarity. We now suppose the natural numbers and \( S \) together satisfy the following five axioms:

1. 0 is a natural number.
2. For every natural number \( n \), we have that \( S n \) is also a natural number.
3. For all natural numbers \( n \) and \( m \), we have that \( n = m \) if and only if \( S n = S m \).
4. For every natural number \( n \), we have \( S n \neq 0 \).
5. Suppose \( P ( n ) \) is a statement about natural numbers. If \( P ( 0 ) \) is true and, for every natural number \( n \), \( P ( n ) \) implies \( P ( S n ) \), then \( P ( n ) \) is true for every natural number \( n \).

I hope you can at least accept the first four of those axioms, we’ll discuss the fifth in a bit more detail later. Hopefully I can convince you to accept that as well if you haven’t already!

As a word of warning, I may drop the word ‘natural’ when talking about the natural numbers, instead just referring to them as numbers. As I’m sure you’ll agree, writing out the word natural over and over again is simply too much effort.

A Definition Diversion

While it is very nice to think that these facts are enough to derive all of number theory, there may, at first glance, seem to be a few glaring omissions. Namely, does the number 1 exist? What is addition? What is multiplication? What other concepts are we missing?

To kick this philosophical discussion off, I’ll throw a definition at you. We will define 1 to be \( S 0 \).

Firstly, does the definition even make sense? Well, axiom 1 tells us that 0 is a natural number, which means it makes sense to talk about \( S 0 \). Secondly, axiom 2 now tells us that \( S 0 \) is a natural number itself. So, to no one’s surprise, we have proven that 1 is a natural number!

At first glance, it might look like we’ve cheated and introduced a sixth axiom, namely \( 1 = S 0 \). This raises the rather subtle point of what mathematicians mean when they provide a ‘definition’.

Imagine that instead of defining 1 to be \( S 0 \), I had defined Peano’s number to be \( S 0 \). I expect that in this case, you would have no problems with accepting this definition—there would be no concern about whether it constitutes an extra axiom. The difference here is that we have a preconception of ‘1’ is, but we don’t have any opinions about what Peano’s number should be.

When provided a definition in formal mathematics, we should throw aside our preconceptions and treat this as something totally new, even if it has a familiar name. In our particular case, when we define \( 1 = S 0 \), we should think of this as a new ‘1’, and not the ‘1’ that we are familiar with.

However, there is a slight hiccup in this line of thinking: how can our axioms of number theory tell us anything about numbers if they require us talk about a ‘new 1’? As we learn more about number theory using the axioms, if our definition of ‘1’ is ‘good’, then we shall find that it behaves exactly how we would expect our ‘preconceived 1’ to behave. We may then claim that anything we learn about our ‘new 1’ actually tells us about the real world, although that is an argument that must be made outside of the framework of mathematics. It is entirely possible that it turns out our definition is ‘bad’ and doesn’t correspond at all with our ‘preconceived 1’; if this turns out to be the case then drawing the distinction between our ‘new 1’ and our ‘preconceived 1’ will have been essential.

With this point cleared, I would like to provide a few more definitions:

\begin{align*} 2 &= SS0 \\ 3 &= SSS0 \\ 4 &= SSSS0 \\ &\cdots \end{align*}

Similarly to how we proved that 1 is a natural number, we can prove that all these new definitions are also natural numbers.

In the next part, we’ll continue our look at the axioms, focusing on the fifth and most subtle of them.