What Are Numbers? Pt. 1: The Natural Numbers

In the last few days in the recent Covid Sydney lockdown period, I had a chance to read and revise on some abstract Algebra concepts such Group Theory, Rings, Fields and Galois Theory. I was reading mainly from the book “Abstract Algebra and Solution by Radicals” by John E. Maxfield and Margaret W. Maxfield amongst other cross-references.

Studying the concepts in Abstract Algebra allow for a greater appreciation for the elementary Algebra presented in High School, provide a clearer understanding of why some things are the way they are (as opposed to just accepting the system of rules), and corrects misconceptions about the theory of Numbers in general.

From the insight I’ve gathered from my readings, I’ve decided to write a series on the number sets that we usually take for granted with more detail, and how they are constructed.

The Natural Numbers are God Given

Let’s start from the beginning with the Natural Numbers. From a young age, these numbers are the easiest to grasp as we can make direct concrete connections between the number of objects we can see and interact with in the real world with the abstract concept of Number. For example, the number 3 is useful to describe 3 lots of any object, like 3 fingers or 3 cows, etc. We shall soon find out everything else is built from the Natural Numbers.

Leopold Kronecker echoes this sentiment as he once said, “God created the natural numbers. All else is the work of man.”

So what are the Natural Numbers?

The 19th Century Mathematician Giuseppe Peano postulated the axioms of the Arithmetic of Natural Numbers. They’ve been updated to the following 5 axioms in modern treatments:

  1. \(0\) is a Natural Number. (Peano’s original formulation started it at \(1\) instead of \(0\) – this has been updated to ensure the set of Natural Numbers has an additive identity, i.e. \(n + 0 = n\) for all Natural Numbers \(n\). Some schools of Mathematics still define the Natural Numbers starting at \(1\). What matters is that there is a starting or beginning number.)
  2. The set of Natural Numbers is equipped with a Successor function \(S: \mathbb{N} \rightarrow \mathbb{N}\) and for every Natural Number \(n\), \(S(n)\) is also a Natural Number.
  3. If \(m \not = n\) then \(S(m) \not = S(n)\). i.e. the Successor function is one-to-one (injective).
  4. For every Natural Number \(n\), we cannot have \(S(n) = 0\). i.e. there is no natural number that has \(0\) as its successor.
  5. Given \(P(k)\) is a statement, if \(P(0)\) is true, and if for every natural number \(n\) \(P(n)\) implies \(P(S(n))\) is true, then \(P(n)\) is true for every natural number. This axiom is called the Principle of Mathematical Induction – it is one of the methods of proving statements that rely on the structure of the Natural Numbers in Mathematics, and is taught in the HSC as a Year 12 Extension I and II topic.

Possible Constructions

In this subsection, I’ll list three ways the Natural Numbers can be constructed that behave according to the Peano Axioms. Of course, there are many other ways to construct the set!

  1. \(\mathbb{N} = \{0,1,2,3,\ldots\}\) is how we usually see this set listed, with a successor function whose rule we can define as \(S(n) = n+1\) for every natural number \(n\).
    We can see that there is no number with \(0\) as its successor, and each successor of a non-zero natural number is found by adding \(1\) to it.
  2. Von Neumann’s Ordinals has the distinguished element the empty set: notated as \(0 = \{\}\). The successor function is \(S(n) = n \cup \{ n \}\), and this gives \[1 = 0 \cup \{0\} = \{\} \cup \{0\} = \{0\}\] \[2 = 1 \cup \{1\} = \{0\} \cup \{1\} = \{0,1\}\] \[3 = 2 \cup \{2\} = \{0,1\} \cup \{2\} = \{0,1,2\}\] etc. This is the definition of Natural Numbers that I personally subscribe to as my head canon.
  3. Ernst Zermelo’s construction is a really interesting one. He defines the set \(0 = \{\}\). The set \(1 = \{0\} = \{\{\}\}\). The set \(2 = \{1\} = \{\{\{\}\}\}\). The set \(n = \{n-1\} = \{\{\{\ldots\}\}\}\). Hence, each natural number is actually a set with this construction! The successor function maps every natural number to the next by containing it in another set.

Algebraic Properties

So what can we do with Natural Numbers?

The most intuitive operations that one can understand concretely is addition and multiplication, but how do they fit in with the description and construction above?

Addition \(+\) can be defined recursively (note that this is possible because of the fifth axiom of the principle of mathematical induction) as follows:

  • Base Case: \(a + 0 = a\)
  • Inductive Step: \(a + S(b) = S(a+b)\).

For example to see how this works:

1+1 & = 1 + S(0) \mbox{ Since $1$ succeeds $0$}\\
& = S(1+0) \mbox{ By our definition}\\
& = S(1) \mbox{ By $a+0=a$ definition}\\
& = 2\\
1+2 & = 1 + S(1)\\
& = S(1+1)\\
& = S(2) \mbox{ From the computation above}\\
& = 3

Multiplication \(\times\) is defined as:

  • Base Case: \(a \times 0 = 0\)
  • Inductive Step: \(a \times S(b) = (a\times b) + a\)

Let’s try this out with \(3 \times 2\):

3\times 2 & = 3 \times S(1)\\
& = (3 \times 1) + 3\\
& = (3 \times S(0)) + 3\\
& = ((3 \times 0) + 3) + 3\\
& = (0 + 3) + 3\\
& = 0 + (3 + 3)\\
& = 3+3

Now we can apply the definition of Addition to get the answer of 6.

1 vs 0?

As hinted at before, some people take the number \(1\) as the starting number and not \(0\). Arguments for or against the inclusion of \(0\) in \(\mathbb{N}\) can get quite lengthy and sometimes even controversial. Instead of going through what those arguments are, I’ll just share how I see it in my personal mathematical framework.

To me, the first element of the Natural Numbers as defined using Vonn Neumann’s construction is the empty set, which can be notated as either \(1\) or \(0\), depending on preference. Although the notation may differ from person to person, logically it is representing the same object – the starting element of the set of Natural Numbers.

Where the notational difference may matter is when we intend to allow the set of Natural Numbers have an additive inverse, i.e.

\[ \forall n \in \mathbb{N} \,\exists {-n} \in \mathbb{N}: n + ({-n}) = 0 \]

If we do allow this algebraic structure, we make \(\mathbb{N}\) into something called a Monoid. If we do not allow this axiom of additive inverse, then we impose upon \(\mathbb{N}\) the structure of a Semi-Group, which is a simpler structure than that of a Monoid.

I guess what I’m saying is, it doesn’t actually matter whether you use \(1\) or \(0\) as the notation for \(\{\}\)… The structure of the set is more important than the notation, or even the heuristic arguments for including or excluding the notation of \(0\).

In the next post, the Natural Numbers act as the building block, or foundation, on which Integers are constructed. We find that we don’t actually need any more axioms on the numbers than the five Peano Axioms to define the Integers. This is amazing, as only five axioms can lead on to the construction of sophisticated sets of Numbers such as the Complex Numbers! Further to this, the axiom about the principle of mathematical induction alone is also given its own topic of study in high school maths courses like Extension I and II!

Part 2: The Integers

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