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From the definition of a sum in § 3 follows:

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