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On account of Theorem 4, we define the function k(n) (n = 1, 2,. 7) . , x,,,+i) Evidently k(n) lies in S. e. ). 7), k(n + 1) = (k(n), f(n + 1). ). 8), we find that f is the required function. Consequently Theorem 7 is proved. The above-mentioned axioms I, II agree with the axioms of Peano, by which he was the first to found the concept of the natural numbers axiomatically. These axioms are 1. 1 is a natural number. 2. To every natural number n there belongs uniquely a natural number n'. 3. For every n, n' 0 1.

This contradiction provesTheorem6. A second kind of recursive definition is contained in the following. THEOREM 7. ) assign a function g,, of n variables xl, . , x,, (= 1, 2,. ) with the range S. f(l) = a. ) is valid. The uniqueness statement of this theorem follows obviously by induction. It still remains to be established that such a function f can be given. ) are pairwise disjoint. We then put S=SUS2U... x E S" holds for every x E S for only one n (= 1, 2,.. ). ,x,, E S). 14 SET-THEORETICAL PRELIMINARIES So if we put for this x h(x) = 9n(x1......

The use of parentheses () and brackets [) in connection with compositions will be discussed here, briefly. First, for a o fi we might have introduced the "more complete notation" (a o 9). g. of the form ((a o ,B) o (y o S)) is then quite clear. For this we write, omitting the outer parentheses, (a o 9) o (y o S). This example is sufficient to illustrate the general case. STRUCTURES 30 In the following, we consider in S the previous multiplication a#. This we call associative, if, for all a, 9, y there always holds (an)y = a(fly) (a, jS, y E S) .