In the past mathematicians wished to created a foundation for all of mathematics. The number system can be constructed hierarchically from the set of natural numbers $\mathbb{N}$. From $\mathbb{N}$, we can construct the integers $\mathbb{Z}$, rationals $\mathbb{Q}$, reals $\mathbb{R}$, complex numbers $\mathbb{C}$, and more. However, it is desirable to be able to construct the naturals ($\mathbb{N}$) from more basic ingredients, since there is no reason $\mathbb{N}$ should itself be fundamental.

The Peano axioms (1889) are a set of axoims that allow for the construction of the natural numbers without ever referencing concepts such as arithmetic or counting. In this way, these axioms are fundamental.

## Background Concepts

One should be familiar with the concept of a set, and that two sets with the same elements means that they are the same set. Secondly, we define a binary operation $=$, known commonly as equals, that is reflexive ($x=x$), symmetric ($x=y \implies y=x$), and transitive ($x=y \wedge y=z \implies x=z$). These may seem obvious, however, they are key for the definition of what we consider equality to hold true. Lastly, we require that the set $\mathbb{N}$ that we wish to construct with these axoims is closed under this $=$ operation. Finally, we require the notion of a map / function. This is simply something that maps inputs to outputs.

## Axoim 1

$a \in \mathbb{N}$

This axiom essentially forces the set under construction to be nonempty: $\mathbb{N} \neq \emptyset$. We state that there is some element $a$ that is a member of our set.

## Axoim 2

$\exists S \ni x \in \mathbb{N} \implies S(x) \in \mathbb{N}$

## Axoim 3

$\nexists x \in \mathbb{N} \ni S(x) = a$

## Axiom 4

$x, y \in \mathbb{N} \wedge S(x) = S(y) \implies x=y$

Here we are essentially stating that our map $S$ is injective.

Axioms 1-4 allow us to define a concept of next or successor without ever explicitly imposing preconceived notions about numbers. Now, if we associate each value of our successor function $S$ with some symbol, it starts looking a lot like the set of natural numbers has been constructed. For example, if we define $0 := a$, $1 := S(a)$, $2 := S(S(a))$, this seems very similar to the natural numbers.

However, we are not done. There is still a loophole that leads to a contradiction. Consider $e_1, e_2 \in \mathbb{N}$, and that $S(e_1) = e_2$ and $S(e_1) = e_1$. This does not violate Axoims 1-4. This somehow allows for a set that seems bigger than $\mathbb{N}$ since $e_1$ and $e_2$ are detached from every other element in $\mathbb{N}$.

## Axiom 5

Suppose $\exists T \subset \mathbb{N}$ such that:

$a \in T \wedge$

and

$x \in T \implies S(x) \in T$

The only such set $T$ is $\mathbb{N}$. This axiom circumvents the above-described loophole.

Please not that this blog post is, in part, a summary of this video.