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By Redei L.

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Let a proposition A" be in unique correspondence with every natural number n. If for every n, the truth of A,, follows from the supposition of the truth of all propositions AX (x < n), then A is true for all it. If A were false for one n, then according to Theorem 5 there would be a least n of this nature. Since then all At (x < n) are true, the truth of A follows from the induction assumption. This contradiction provesTheorem6. A second kind of recursive definition is contained in the following.

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...... 6) then a function h(x) (x E S) with range S is defined. 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.

The most important of these structures are the groups, rings and skew fields (in particular fields). Accordingly, the most important chapters of algebra are group theory, ring theory and the theory of skew fields (in particular field theory). It is not easy, however, to define sharply the limits of these chapters, as they are so intermingled and interdependent and, consequently, they cannot successfully be developed as independent theories. STRUCTURES 37 When we speak of groups, then of course modules are included, since these are themselves special groups.

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