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.