By Abraham A. Fraenkel

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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). The paradox appearing in this contrast, clearly pointed out already by Galileo l ) , has fulfilled an important but decisively negative task in the history of the conquest of infinite magnitude for the realms of mathematics and philosophy : the infinite aggregates, having so paradoxical a quality, seemed to be discredited.

Now, all our operations are indeed justified. The existence of the sum introduced through definition 111, is guaranteed by the principle of sum-set ”. As t o the existence of the meet, let S, be any 3, of the given sets, according t o definition 111. For every element x of XI the property “ x is contained in each of the given sets S,” is meaningful. The elements x of S, that possess this property form a set, according t o the principle of subsets; more exactly, they form a certain subset of S, which is precisely the meet of the setJs S,.

I k , a certain set 1) of the collection will correspond while no other sets appear in the collection. It is not desirable to include the first case in the second since a sequence, according to our explanation, implies a certain order in which the sets of the collection are given. We thus define: DEFINITION111. Given a finite number of sets S,, S,, . . )S,, or a sequence 2) of sets S,,S,,S,,. . , S k , .. , by their sum (join) or sum-set J we understand the set of all elements contained in at least one of the sets S,, and by their meet or inner product 3) M , the set of all elements contained in each of the sets S,.