By Abraham A. Fraenkel

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**Example text**

We start with an assertion that is almost self-evident: THEOREM 5. A set which is equivalent t o a finite (infinite)set, is again finite (infinite). Proof. The proof based on definition \'I is left to the reader. We shall base our proof on definition V I I a,nd show that a set T which is equivalent t o a reflexive set X, is again reflexive. S being reflexive, a proper subset of it, S ' , exists such that X 8'. Since T - S , there exists a representation 9 between T and 8, to which we shall keep.

We saw that these sets differ from each other only with regard to the nature of their elements; their elements can be related to (attached t o ; paired with) each This condition is really superfluous. For those elements of A which l) are not sets, are automatically eliminated by the formation of the sum-set, since they do not contain any elements. Nevertheless, the condition may psychologically facilitate the understanding of the concept sum-set. z, I n the sequence mentioned in definition 111, the same set may, of course, appear several times.

Arrange the fractions corresponding to a definite value of s in the order mentioned before, and put them after the fractions corresponding to smaller values of s. Finally, in order to include rationals other than positive, put O / l = 0 (corresponding to s = 1) a t the beginning of the entire sequence, and let every fraction mjn be followed by the negative fraction - m/n. Thus we obtain a sequence of different rational numbers I) beginning with 1 1 1 2 2 1 0; 7’ - i ; 7, - 7 , 2, 5 -5 1 1’ 1’ 5’ 1 6 6 5 .

### Abstract Set Theory by Abraham A. Fraenkel

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