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By Abraham A. Fraenkel

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Example text

Meanwhile it is sufficient to generalize in two special directions: first from two sets to any finite number of sets (cf. e. , the property of being different both from l) a and from b. Or, of differing from b. ). I n Latin, however, one has two different particles: aut (exclusive), and we1 (at least one of the two). Compare the sentences “this day is Monday or ( a u t ) Thursday” and “in the next or (wel) the third block you will 6nd a taxi”. T h e “or” of logical disjunction, sometimes called “alternation”, is wel.

I n Foundations, ch. 11, various methods of defining finiteness and infinity will be compared in the light of that principle. Cf. also 0 10, 6 . 42 E L E M E N T S . C O N C E P T O F CkRDINAL NUMBER [CH I sl,s2, . , s,,, . .. L r t be S - S* = S’ (S’= 0 not excluded) so t h a t = N* +A’‘, S* and S ’ being exclusive. e. the mate of 1 ) from S*, u e obtain the set AS: = {s,, ss, . , sk, . . }. Let be AS:‘ S’ = So. So c S IS also true; in fact, So does not contain the element sl. S $ and S’ are again exclusibe.

Beginners who want to dispute the possibility of a set being equivalent to a proper subset, frequently make the mistake of basing their arguments on the identical correspondence as though it were superior to other correspondences. Many other instances of sets being equivalent although the one is a proper subset of the other, will appear in $$ 3 and 4. 40 ELEMENTS. C O N C E P T O F CARDIKFAL XUMBER [CH. I The phenomenon that a set can, as it were, be of the same extent as a proper subset, stands in some contrast to the old principle toturrb parte rnaius (the whole is larger than a part).

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Abstract Set Theory by Abraham A. Fraenkel


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