## 8.1.Natural Numbers

The natural numbers, in the sense used here, are the numbers 0, 1, 2, 3, ... This is the predominant definition in CS.

In number theory, there is only one special symbol 0. All other numbers are by means of the successor mechanism. The successor of n, written as s(n), is merely the number following n in the sequence of natural numbers.

```1.    =   s(0)
2.    =   s(1) = s( s(0) )
3.    =   s(2) = s( s(1) ) = s( s( s(0) )
...
```

There are a numbers of axioms that describe how to work with natural numbers. The axioms are introduced by Guiseppe Peano and they are therefore called Peano axioms.

Peano Giuseppe Peano; born: 27 Aug 1858 in Cuneo, Piemonte, Italy. Died: 20 April 1932 in Turin.

Peano was the founder of symbolic logic and his interests centered on the foundations of mathematics and on the development of a formal logical language.

Peano studied mathematics at the University of Turin and joined the staff there in 1880. In 1889 Peano published his famous axioms, called Peano axioms, which defined the natural numbers in terms of sets.

He produced an axiomatic definition of the natural number system and showed how the real number system can be derived from these postulates.