As a reminder, here are Peano’s axioms, which concern the successor function , which intuitively “adds ” to each natural integer.
These axioms allow us to reconstruct the whole arithmetic structure of the set of natural integers, and from naive set theory, all natural mathematical objects and structures.
is not the successor of any natural number. In other words, there is no natural number such that .
This axiom says in particular that the function is not surjective, since the number has no antecedent by .
If two natural numbers have the same successor, then they are equal. In other words, for all natural numbers , if then .
We can reformulate this axiom by saying that the successor application is injective. It thus defines a bijection from onto . We recognise here the characterisation of an infinite set.
The set of natural numbers is infinite.
Axiom 3 [Principle of induction (or recursion)]
If is a subset of such that:
- for all , (“induction step”),
then we have .
This principle expresses intuitively that the set is entirely \og browsed” if we enumerate it starting from and add each successive natural number, indefinitely.
We can deduce the determination of the image of :
If we admit the induction principle, then the image of is , the set of non-zero natural numbers.
Let be the set . If we show that , then any non-zero natural number is in the image of , so . By definition, we have , and suppose that is a natural number, and that . By definition, the natural integer is in ! By the induction principle, the set is the whole set , and the proposition is proved.