By Abraham A. Fraenkel

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For any natural number k . The sum is the set of all natural numbers, coinciding therefore with X,. The meet is the null-set. e. any element of S,; then n is not contained in the set and therefore not in the meet. o different natural numbers. In other words : I n A srquenrcl - in corit,rast, to a set, cf. p. 17 - the same element may appmr several times. For the connection between tkle concepts of set and of sequence, see 4. g. (,TI, s,, s,,. ). Since sets are tlcnoted by curly brackets { }, no confusion will arise between a set and it sequence, if eit’her is denoted by its members.

C‘ompare also exercise 5 at the end of this section. For a rcrtain genrritlization of the equivalence relation, see Tola 1. 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.

I 2n-1 I 2% ... CH. I, $ 31 DENUMERABLE SETS 47 By this procedure the set M has been enumerated; it is, therefore, a denumerable set. Evidently one does not alter the denumerability of M by adding the element 0 ; in general the denumerability of an aggregate is not changed by the addition of a finite number ( k ) o f new elements. One may, for example, put the new elements a t the beginning of the new enumeration, and the only change resulting from this will be an increase of the index which assigns to each element its place in the sequence; in our case, an increase by the constant value k .

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