Lesson 4

The incredible success of Euclid's axioms leads to a natural question: Can we come up with a nice set of axioms for more general concepts in mathematics?

In the late nineteenth century, mathematicians started to realize that this question was fundamental to their field. For example, the set of natural numbers \(\mathbb{N}\) can be defined as follows. It is known as Peano axiomatization and uses the following non-logical symbols: a constant \(0\) and a unary function symbol \(S\) (standing for successor).

1. Zero is a natural number: \(0 \in \mathbb{N}\)
2. Equality is reflexive: \(\forall x \in \mathbb{N}\colon x=x\)
3. Equality is symmetric: \(\forall x,y \in \mathbb{N}\colon x=y \Rightarrow y=x\)
4. Equality is transitive: \(\forall x,y,z \in \mathbb{N}\colon x=y \land y=z \Rightarrow x=z\)
5. \(\mathbb{N}\) is closed under equality: \(\forall a, b \colon b \in \mathbb{N}\land a=b \Rightarrow a \in \mathbb{N}\)
6. \(\mathbb{N}\) is closed under \(S\): \(\forall n \in \mathbb{N}\colon S(n) \in \mathbb{N}\)
7. \(S\) is an injection: \(\forall n,m \in \mathbb{N}\colon S(n)=S(m) \Rightarrow n=m\)
8. \(0\) is not a successor of anything: \(\forall n \in \mathbb{N}\colon S(n) \neq 0\)
9. Induction: \(\forall P \colon \left( (P(0) \land (\forall n \in \mathbb{N}\colon P(n) \Rightarrow P(n+1))) \Rightarrow (\forall n \in \mathbb{N}\colon P(n)) \right)\)

Again, the last axiom, induction, is essential as there are models where it does not hold.