Page:Cantortransfinite.djvu/114

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OF TRANSFINITE NUMBERS
95

for all 's which are different from .

The totality of different coverings of N with M forms a definite aggregate with the elements ; we call it the "covering-aggregate (Belegungsmenge) of with " and denote it by . Thus:

(2)
.

If and , we easily find that

(3)
.

Thus the cardinal number of depends only on the cardinal numbers and ; it serves us for the definition of  :

(4)
.

For any three aggregates, , we easily prove the theorems:

(5)
,
(6)
,
(7)
,

from which, if we put , we have, by (4) and by paying attention to § 3, the theorems for any three cardinal numbers, , , and :

(8)
,
(9)
,
(10)
.