(D2.1.1): A natural number is any element of the set:
Let \(n++\) denote the successor to \(n\).
So really, let \(\mathbf{N}\) be the number 0 and all its successors.
What is 0? We define it axiomatically.
(Axiom 2.1): \(0\) is a natural number.
(Axiom 2.2) If \(n\) is a natural number, then so is \(n++\).
(D2.1.3): Let 1 be \(0++\), and 2 be \(1++\), and so on.
(P2.1.4) \(3\) is a natural number.
(Axiom 2.3): 0 is not the successor of any natural number.
This prevents the numbers from cycling back to the start.
(P2.1.6): 4, which is the successor of 3, is not 0.
(Axiom 2.4): Different natural numbers have different successors. Thus, if \(n\ne m\) then \(n++\ne m++\) Also, if \(n++=m++\), then \(n=m\).
This prevents smaller cycles. It also guarantees that the numbers go on forever.
P(2.1.8): 6 is not equal to 2. This can be shown because 6 is a successor of 5 which is a successor of 4. Likewise 2 is a successor of 1 which is a successor of 0. And we know \(4\ne 0\). Thus \(5\ne 1\) and \(6\ne 2\).
(Axiom 2.5): The principle of mathematical induction. Let \(P(n)\) by a property pertaining to the natural number \(n\). Suppose that \(P(0)\) is true, and suppose that whenever \(P(n)\) is true, so is \(P(n++)\). Then \(P(n)\) is true for all natural numbers.
This kind of precludes the possibility of having numbers that are not “reachable” from 0.
The above 5 axioms are called the Peano axioms.
(P2.1.16) Suppose for each natural number \(n\), we have a function \(f_{n}: \mathbf{N} \rightarrow \mathbf{N}\). Let \(c\) be a natural number. Then we can assign a unique natural number \(a_{n}\) to each natural number \(n\) such that \(a_{0}=c\) and \(a_{n++}=f_{n}(a_{n})\) for each natural number \(n\).
Informally we want to show that for every \(n\), there exists an \(a_{n}\) and \(a_{n}\) will have only one value.
Proof:
Use induction. Assign \(a_{0}=c\). Because of Axiom 2.3, none of the other definitions will define \(a_{0}\). Now given \(a_{n}\), show that \(a_{n++}\) is well defined (i.e. can have only one value), using Axiom 2.4.