Creating The Natural Numbers! (peano`s Axioms)

Note, the article will be quite advanced. The concept of Peano Axioms are usually taught in university in discrete maths courses.

Peano Axioms

Motivations and History

During the late nineteenth century, there was a drive to "formalise" mathematics, which took place through the axiomatisation of many fields. Axiomatisation is when we reduce a concept down to its basic fundamental truths (known as axioms) that we accept without proof. The intuition is that we need something at the bottom of mathematics from which we can reason upwards.

The Peano Axioms came across from the axiomatisation of the natural numbers they are five axioms describing the natural numbers. These axioms are declarations that are unprovable, and we accept/assume they are true.

There are many other axioms, https://en.wikipedia.org/wiki/List\_of\_axioms. Note that not all concepts need to be described directly with axioms. Concepts such as integers can be defined using the definition of natural numbers, and the concept of rational numbers can be defined using the definition of integers.

The Five Axioms

Rather than directly stating the axioms, I will take you on a journey on how we can axiomatise natural numbers.

Firstly, we need to understand what natural numbers are. They are the numbers that occur commonly and obviously in nature, such as 0, 1, 2, 3, ... Different definitions of natural numbers either include or exclude 0, though we usually include it when describing the Peano axioms for the sake of simplified arithmetic. Though when Peano originally wrote the axioms, he described the natural numbers starting at 1!

We will be using two symbols a constant 0 and a unary (takes one input) function Succ, known as the successor function. In other descri ptions of the Peano Axioms, you may see Succ(x) written as Successor(x) or shortened to S(x) or even in a programming style as x++. Though the naming does not matter and is there to describe the intuitive sense that Succ(x) succeeds (is larger than) x. Though the actual behaviour of the function is described through the axioms.

We need a starting point to build the natural numbers, so we introduce 0 into the natural numbers.

1st Axiom: $0$ is a natural number

We still need to describe the other natural numbers and want to say Succ(0), Succ(Succ(0)), Succ(Succ(Succ(0))), ... are also natural numbers. To do this, we can build off 0, with the axiom:

2nd Axiom: If n is a natural number, then Succ(n) is also a natural number.

Though there is a problem with this, let us take two natural numbers y and z. Let`s assume y = z, then it makes sense for Succ(y) = Succ(z). It`s like saying if 4 = 4 then 4 + 1 = 4 + 1, however the current rules have a loophole. Just because y = z, does not mean that Succ(y) = Succ(z), they can be completely different, and that contradicts the meaning of the natural numbers that we are trying to describe. Note, the same problem occurs the other way around just because Succ(y) = Succ(z) does not mean y = z. So we introduce the third axiom to fix this issue:

3rd Axiom: For all natural numbers n and m, m = n if and only if Succ(m) = Succ(n).

We have fixed one problem, but we spot another issue. Let`s take some arbitrary number x, currently the rules allow Succ(x) = 0. Notice a problem? Succ(x) = 0 is similar to saying there are numbers x before 0, which contradicts the meaning of the natural numbers (the natural numbers are all above 0). So we introduce the 4th axiom. Notice that we are introducing these axioms out of necessity for a purpose, to describe what the natural numbers are as simple as possible.

4th Axiom: For every natural number n, Succ(n) = 0 is false.

We have said what is in the natural numbers, but we have not said what is not. We have not said that the natural number does not contain other elements such as 3.1415... or a dog! We want to say what the entirety of the natural numbers are and what it is limited to. So we introduce the fifth and final axiom, which is also known as the principle of induction:

5th Axiom: Making a set K such that, if 0 is in set K and if for any natural number n in set K, Succ(n) is in set K. Then set K contains every natural number.

Lastly, we can assign symbols which are familiar to us, to these numbers, to abstract and manipulate them with greater ease.

0 to 0

Succ(0) to 1

Succ(Succ(0)) to 2

Succ(Succ(Succ(0))) to 3

etc...