By Abraham A. Fraenkel

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**Example text**

Then the meet of the sets of points included in the polygons is the set of points inside the circle and on its circumference. The following remark, ensuing from examples 1) and 2), will prove useful in 5 5 : Given a sequence of sets such that a n y set i s a subset of the preceding set of the sequence, the meet of all sets m a y either be the null-set or diger from it. By the way, we again notice the expediency of definition I1 on p. 2 3 : had we not introduced the null-set, we would not be able to maintain that the meet of any sequence (or set) of sets is again a set.

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). 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.

Otherwise we should have as the remainder a finite set, which is not “denumerable” 2 ) . e. n t o I 1”= n = 9 f 2n; or in a scheme ; p = j. 1 2 3 4 ... y k ... k ... The general case is aJgain provided for by the procedure described in the last paragraph : by arranging the remaining elements according to their magnitude. Accordingly, any infinite (non-inductive) subset o€ the set of all natural numbers is again denumerable. I n reaching this result we have not used any particuhr property of the natural numbers (besides that expressed in footnote 1).