By Abraham A. Fraenkel

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For our exposition it is easily settled by the principle of infinity (at the end of $ 2 ) which postulates the unconditional existence of a set. Strictly speaking, the first sentence in theorem 2 should read: The null-set exists if there exists a set a t all. CH. I, $ 21 THE FUNDAMENTAL CONCEPTS. FINITE AND INFINITE 25 the pair P = {a, b} exists. Take the property n of not being a n element of P I ) ; by principle 11, n creates from P the null-set. e. the set {a}. D. It is easy to perceive how this proof may be generalized in order to prove the existence of any finite subset of a given set.

Sn} of S is equivalent to a proper subset of itself - which contradicts our assumption. We shall consider three possibilities : a) The subset S' does not contain the element s , + ~ of S. Then the mate x of s , + ~ E S in S' is different from Let us write S' = S" { x }where S" is a proper subset of S' ,). Therefore, when we remove sn+lE S and its mate z ES' from the supposed representation between S and S', there remains a representation between {s,, s,, . . , sn} and its proper subset S", contrary t o our assumption that for a set of n elements no such representation exists.

CH. I, § 21 T H E FUNDAMENTAL CONCEPTS. FINITE AND INFINITE 37 equivalence in the general sense. The proof may be left to the reader. Finally, let us remark: From the definition of representation it immediately follows that a given representation between the (equivalent) sets S and T represents a n y proper subset of S upon a proper subset of T . Likewise, if S, and S, are mutually exclusive sets (p. 20), and if v1 represents S, upon T,, while y , represents S, upon a set T , (exclusive to T,),then the set S, + S, i s represented u p o n T,+ T , by the join of tlze representations yr and q2,and this join again is a representation.

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