Succession of cardinal-numbers

I now come to the promised proof that the cardinal-numbers of (I) and (II) follow upon one another directly, so that there are no other cardinal-numbers in between. If one chooses a set (α′ ) of different numbers α′ from the number-class (II) according to some rule, i. e., any set (α′ ) contained within (II), then such a set always has peculiarities, which can be expressed in the following sentences:

“There is always a smallest number in the numbers of the set (α′ ).”

and

“If, specifically, one has a sequence of numbers of the number-class (II): α1, α2, …, αβ … which continuously decrease in size (so that αβ > αβ′ if β > β′ ), then this sequence necessarily terminates with a finite number of elements and ends with the smallest of the numbers; the sequence cannot be infinite.”

It is remarkable that this proposition, which is immediately clear when the numbers are finite integers αβ can also be demonstrated in the case of infinite numbers αβ . Indeed, according to the previous proposition, which can easily be seen from the definition of the sequence of numbers (II), there is a smallest number among the numbers αv if one only considers those ones for which the index v is finite; if for some ρ this number is = αρ, then it is evident that αv > αv + 1 and the sequence αv and thus also the entire sequence αβ must consist exactly of ρ elements, and so it is a finite sequence.

Now one obtains the following fundamental proposition: “If (α′) is any set of numbers contained in the number-class (II), only the following three cases can occur:

(α′ ) is a finite set, i.e: it consists of a finite number of numbers, or (α′ ) has the infinite cardinal-number of the first class, or (α′ ) has the infinite cardinal-number of (II).” Quartum non datur (The fourth is not given).

The proof can be carried out as follows: let Ω be the first number of the third number-class (III); then, because the latter is contained in (II), all the numbers α′ of the set (α′ ) are smaller than Ω.

We now think of the numbers α′ in order of their size; αω is the smallest among them, αω + 1 the next largest, etc., so that the set (α′ ) is obtained in the form of a “well-ordered” set αβ, where β runs through numbers of our natural extended number sequence from ω on. Obviously β always remains less than or equal to αβ and since αβ < ω, then β < ω. The number β cannot therefore go beyond the number-class (II), but remains within its domain; therefore only three cases can take place:

β remains below a specific number in the sequence w + v, in which case (α′ ) is a finite set, or β assumes all values of the sequence w + v, but remains below a specific number of sequence (II), in which case (α′ ) is obviously a set of the first cardinal-number, or β assumes arbitrarily large values in (II), in which case β runs through all the numbers in (II); in this case the set (αβ), i.e: the set (α′ ) has evidently the cardinal-number of (II). so the proposition is proved.

As a direct result of the theorem just proved, the following apply:

“If one has some well-defined set M of the cardinal-number of the number-class (II) and takes some infinite subset M′ of M, the set M′ can either be thought of in the form of a simply infinite sequence, or it is possible to map both sets M′ and M to correspond one-to-one to each other.”

“If one has any well-defined set M of the cardinal-number of the number-class (II), a subset M′ of M and a subset M′′ of M′ and if one knows that M′′ can be mapped one-to-one to M, then M′ can be mapped one-to-one to M, and also to M′′ .”

I make this last assertion here because of the connection with the preceding ones, given the condition that M has the cardinal-number of (II); it is also obviously correct if M has the cardinal-number of (I). This appears to me to be most remarkable, and I therefore emphasize this expressly, that this proposition has general validity, regardless of the cardinal-number of the set M. I will go into this in more detail in a later article and then demonstrate the peculiar interest which is attached to this general proposition.