Page:Cantortransfinite.djvu/118

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OF TRANSFINITE NUMBERS
99

From the definition of a sum in § 3 follows:

(6)
;

that is to say, every cardinal number, except 1, is the sum of the immediately preceding one and 1.

Now, the following three theorems come into the foreground: A. The terms of the unlimited series of finite cardinal numbers

are all different from one another (that is to say, the condition of equivalence established in § 1 is not fulfilled for the corresponding aggregates).

[490] B. Every one of these numbers is greater than the preceding ones and less than the following ones (§ 2).

C. There are no cardinal numbers which, in magnitude, lie between two consecutive numbers and (§ 2).

We make the proofs of these theorems rest on the two following ones, D and E. We shall, then, in the next place, give the latter theorems rigid proofs.

D. If is an aggregate such that it is of equal power with none of its parts, then the aggregate which arises from by the addition of a single new element , has the same property of being of equal power with none of its parts.

E. If is an aggregate with the finite cardinal number , and is any part of , the cardinal