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Rh 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 $1, 2, 3, ..., \nu,...$ 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 $$\nu$$ 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 $$\nu$$ and $$\nu+1$$ (§ 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 $$M$$ is an aggregate such that it is of equal power with none of its parts, then the aggregate $$(M, e)$$ which arises from $$M$$ by the addition of a single new element $$e$$, has the same property of being of equal power with none of its parts.

E. If $$N$$ is an aggregate with the finite cardinal number $$\nu$$, and $$N_1$$ is any part of $$N$$, the cardinal