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 ˆrst semiaxiomatic presentation of this subject was given by Dedekind in 1879 and, in a slightly modiˆed form, has come to be known as Peano’s postulates.* It can be formulated as follows:
(P1) 0 is a natural number.†
(P2) If x is a natural number, there is another natural number denoted by x′ (and called the successor of x).‡
(P3) 0 ≠ x′ for every natural number x. (P4) If x′ = y′, then x = y. (P5) If Q is a property that may or may 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 x′ has the property Q, then all natural numbers have the property Q (mathematical induction principle).
