The set of natural numbers \(\mathbb{N}\) is defined as follows:¹

1. \(0 \in \mathbb{N}\).
2. \(\forall x. x\in \mathbb{N} \to s(x) \in \mathbb{N}\).
3. \(\neg\exists x\in\mathbb{N}. s(x) = 0\).
4. \(\forall x,y\in\mathbb{N}. s(x) = s(y) \to x = y\).
5. Let \(\phi\) be a unary predicate:

\[\begin{align*} [\phi(0)\land \forall x\in\mathbb{N}. \phi(x)\to\phi(s(x))] \to \forall y\in \mathbb{N}.\phi(y) \end{align*}\]

1. This is a version of Peano Axioms that omits the axioms on equality and differs from the original by taking 0, rather than 1, as the base case. ↩