Peanos Axioms

Natural numbers are what most humans learn to count with in life and in mathematics, we axiomatizing it to create a proper firm foundation for it. In mathematics, Giuseppe Peano is the one credited to have come up with the modern axiomatization of natural numbers. What he wrote is however different from what we use today and I will go through the modern way of it and demonstrate how this axiomatization can be done by ZFC as well.


These are the axioms and we let \mathbb{N} be the set we work with and call the set of natural numbers and the elements are called natural numbers.

  1. \exists 0\in\mathbb{N}
  2. \exists \sigma:\mathbb{N}\hookrightarrow\mathbb{N}/\{0\}
  3. \forall S\forall x((0\in S \land (x\in S\implies \sigma(x)\in S))\implies S=\mathbb{N})

This is amongst the most succinct ways of writing peanos axioms, it can be read out as

  1. There exists one element we call 0 in our natural numbers
  2. We have a function \sigma that is injective but not surjective to \mathbb{N}. We call this function the successor function.
  3. The induction principle which is that if a subset of \mathbb{N} contains 0 and all the successors of its elements, then it is equal to the natural numbers.

It should be noted that some authors choose to start at 1, I generally prefer starting at 0 cause it makes natural numbers into a semiring. We label the elements according to our intuitive feeling, we know that “after 0 comes 1”, and as such, we label 1=\sigma(0), we label 2=\sigma(1)=\sigma^2(0) and so on. Some might question my usage of \sigma^2, where I technically use 2 before I even define it, but again, this is using our intuitive understanding to facilitate how to convey what is meant, I could just as well write \sigma(\sigma(0)) or \sigma\circ\sigma(0) to make it equal but for large amount of it, such that it would later require, it is cumbersome and entirely pointless so we may very well use that intuitive shorthand to make it easier for our human brains.

From these axioms we can show that the structure (\mathbb{N},+,\cdot) has all the properties, if we define addition and multiplication as follows.


Addition

We define addition recursively as following

  • a+0=a
  • a+\sigma(b)=\sigma(a+b)

What this means is that we define it so that our object 0 is the neutral element for addition on the right side, notice we do not define it to be for the left side and as such it might not be, however we will see that it is commutative so it is for the left side as well. After that we say that the sum of an element and the successor of another element, is the successor of the sum of the elements.


Multiplication

For multiplication, we define it also recursively as

  • a\cdot 0 = 0
  • a\cdot\sigma(b)=a+a\cdot b

The recursive step of a\cdot\sigma(b)=a+a\cdot b is the one that captures our intuitive feeling of what natural number multiplication is, repeated addition. It will turn any product we want into a long sequence of additions that terminates at 0 always. A quick proof we can see is that

a\cdot\sigma(0)=a+a\cdot 0 = a+0=a

That is, the successor of 0 is the identity of multiplication.

With this we can now show that for example we have the operation being associative, that is a+(b+c)=(a+b)+c. The sources below demonstrates how it is done and a crucial thing of importance is that they all rely on the induction principle. Going through it all


ZFC

Peanos axioms can be embedded into ZFC by using what it provides us, the definition of addition and multiplication gives functions for us based on the successor function so that is the one we only need to focus on, and the existence of a set that sates the desired qualities and luckily for us, this is not difficult.

We define the successor function as

\sigma(x)=x\cup\{x\}

For a set where this would work is provided by Axiom of Inifnity, namely

\exists S\forall x (\emptyset\in S\land(x\in S\implies x\cup\{x\}\in S))

This set, has the defined successor function built into it and gives us a natural zero element, namely the empty set there would be the element we label zero in peanos axioms. So if we let \mathbb{N}=S and 0=\emptyset then we have \sigma:S\hookrightarrow S/\{\emptyset\} and from there, the rest of it follows naturally from ZFC. This illustrates the power of ZFC, that given an axiomatization of natural numbers, we can use ZFC to construct it instead.

An important thing to pay attention to here is that we are not saying numbers are sets but that we can use sets in ZFC to form a structure that has the characteristics of what we expect the natural numbers to have.


Sources

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