ABSTRACT
Together with geometry, the theory of numbers is the most immediately intuitive of all branches of mathematics. It is not surprising, then, that attempts to formalize mathematics and to establish a rigorous foundation for mathematics should begin with number theory. The first semiaxiomatic presentation of this subject was given by Dedekind in 1879 and, in a slightly modified form, has come to be known as Peano’s postulates.* It can be formulated as follows:
(P1) 0 is a natural number.y
(P2) If x is a natural number, there is another natural number denoted by x0
(and called the successor of x).z
(P3) 0 6¼ x0 for every natural number x. (P4) If x0 ¼ y0, then x¼ y. (P5) If Q is a property that may ormay not hold for any given natural number,
and if (I) 0 has the property Q and (II) whenever a natural number x has the property Q, then x0 has the property Q, then all natural numbers have the property Q (mathematical induction principle).
