11.3 Models for \(\mathbb N\)

We now see that the very fundamental axioms of Peano, together with the (not so simple?) definitions of addition, multiplication and order give rise to a set with the same properties we would expect the natural numbers to satisfy. It still doesn’t really answer the question of what the natural numbers actually are but at least it moves us a step further along that road.

Suppose that we find a set \(N\) together with a function \(t:N\to N\) such that

1. \(t\) is an injection;
2. \(t\) is not a surjection - more explicitly, there exists \(z\in N\) such that \(z\not\in\text{ im }(t)\);
3. If there is a subset \(A\subseteq N\) with the property that \[ z\in A\text{ and }t(x)\in A\text{ whenever }x\in A \] then \(A=N\).

It should be clear by a similar argument to that used for the system \((\mathbb N,s,0)\) that \[ N=\{z, t(z), t(t(z)),\ldots\}. \] It is also clear that we can define a function \(b:N\to\mathbb N\) by \(b(z)=0\) and \(b(t(x))=s(b(x))\) and you should check that \(b\) is a bijection which preserves the successor functions and the ‘zero element’. We say that \(N\) is isomorphic to \(\mathbb N\) (or more correctly that \((N,t,z)\) is isomorphic to \((\mathbb N,s,0)\)).

We say that a set \(N\) together with a function \(t:N\to N\) and an element \(z\in N\) which satisfy the above conditions is a model for the natural numbers. Our previous argument essentially says that, all models for \(\mathbb N\) are isomorphic. Consequently, if we can describe a particular model, then we may wish to treat that particular model as a ‘candidate’ for an answer to the question ‘What is the natural numbers?’.