- (Pe1)
- There exists an injection
. For
, is called the
*successor*of . - (Pe2)
- There exists an element, called
*zero*and denoted by 0, which has no pre-image by . - (Pe3)
- Principle of Induction: If a subset of contains 0 and the successor of each of its elements, then this subset is equal to .

- As is an injection, two different natural numbers have different successors.
- Denote
; then the first axiom says:

- In order to be consequent with the intuition (and the ordinary notations), we have: , , etc.
- There exists no element such that .
- The principle of induction says that if a property is true for 0 and is
*hereditary*, i.e. passes from a number to its successor, than this property is true for all the natural numbers. For example, people can infect one the other with some illness, but if nobody suffers from this affection, then the epidemy has no possibility to begin.

''either the number is equal to 0 or it has a predecessor''

.
- The fact that is obvious.
- Suppose that some natural number belongs to . Then has a predecessor, thus .

- (i)
- holds for the natural number .
- (ii)
- If holds for a natural number , then holds for .

**Solution**- By induction on .
- For , the polygon is a triangle; the sum of its angles is .
- Suppose that for some natural number , the desired property holds. Consider a convex polygon with vertices and denote the vertices clockwise by , , , . By the induction hypothesis, the sum of the angles of the convex polygon is equal to ; moreover, the sum of the angles of the triangle is . Thus, the sum of the angles of the polygon is .