# A Foundation of Mathematics - The Peano Axioms

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.