4 Natural numbers[1X9]
We want to properly define the set
\[ {\mathbb N}=\{ 0,1,2,\ldots \} \]
of the natural numbers.
A possible model, as shown in Sec. [246], is obtained by relying on the theory of Zermelo—Fraenkel.
Here instead we present Peano’s axioms, expressed using the naive version of set theory.
Definition
295
Peano’s axioms
From those two important properties immediately follow. One is the principle of induction, see [1XC]. The other is left for exercise.
Exercise
296
The idea is that the successor function encodes the usual numbers according to the scheme
\[ 1=S(0),\quad 2=S(1), \quad 3=S(2)\ldots \]
and (having defined the addition) we will have that \(S(n)=n+1\).
Exercise
297