- There exists a natural number zero (0).
- For each $ n\in\N $ , there exists a natural number that is the successor of $ n $ , denoted by $ n' $ .
- 0 is not the successor of any natural number. (For each $ n\in\N,n'\ne0 $)
- For all natural numbers $ n,m\in\N $ , if $ n'=m' $ , then $ n=m $ .
- Given any predicate $ P $ on the natural numbers, if $ P(0) $ is true and $ P(k) $ implies $ P(k') $ for any $ k\in\N $ , then $ P(n) $ is true for all $ n\in\N $ . (This is also known as the principle of mathematical induction.)
A variation of these axioms substitute the natural number one (1) in place of zero in axioms 1, 3 and 5 above. It is possible, therefore, to regard the natural numbers as excluding zero while the whole numbers include zero. Usage varies on this point, however.
The Peano axioms set up the natural numbers by introducing a particular element (zero) with the first axiom, and then describing a particular function on the natural numbers. This function, namely the successor function, sets up a chain that begins at zero (axiom 3), and continues indefinitely (axiom 2). Axiom 5 ensures that the chain will eventually cover every natural number, and axiom 4 ensures that every link in the chain is preceded by at most one other link.
Intuitively, there are more properties of the natural numbers than are listed in the axioms. However, these properties can be proven from the axioms. Here are some such properties:
- No natural number is its own successor
- Every natural number except for zero is a successor to some other natural number