Cardinal-numbers

The first thing we now have to show is the theorem that the new number-class (II) has a cardinal-number that is different from that of the first number-class (I).

This proposition results from the following proposition:

“If α1, α1, …, αv, … is any set of distinct numbers of the second number-class where that set is of the first infinite cardinal-number (so we can assume that they are in the form of a simple sequence form (αv)), then either:

one of them is the largest, which we call γ, or there is a certain number β of the second number-class (II) that does not appear among the numbers αv, such that either β is greater than all αv, or alternatively stated, every integer β ′ in the second number-class, where β ′ < β, is exceeded in size by certain numbers in the sequence (αv). The numbers γ or β respectively can correctly be called the upper limit of the set (αv).”

The proof of this proposition is simply as follows: In the sequence (αv) let αx2 be the first number which is greater than α1, and αx3 the first number which is greater than αx2 and so on.

One then has:

1 < x2 < x3 < x4 < …

α1 < αx2 < αx3 < αx4 < …

and

α v < αxλ

whenever

v < xλ

Here it can occur that from a certain number αxρ on, all numbers in the sequence (αv) that follow it are smaller than it; then it is obviously the largest of them all and we have: γ = αxρ . Otherwise, consider the set of all integers from 1 onwards that are smaller than α1, and first add to this set, the set of all integers which are ≥ α1 and < αx2, then the set of all numbers which are ≥ αx2 and < αx3 and so on, one then obtains a definitive subset of consecutive numbers of our first two number-classes. This set of numbers most obviously is of the first infinite cardinal-number, and therefore there exists (according to the definition of the second number-class (II)) a certain number β of the number-class (II), which is the next largest to those numbers, and so we have β > α xλ and therefore also β > αv because xλ can always be assumed to be so large that it becomes larger than a given v and because then αv < α xλ .

On the other hand, it is easy to see that every number β ′ < β is surpassed in size by certain numbers α xv , by which all parts of the theorem are now proved.

From this follows the proposition that the set of all numbers of the second number-class (II) does not have the cardinal-number of the first number-class (I); because otherwise we would get the integer number-class (II) in the form of a simple sequence:

α1, α2, …, αv …

and so it must have, according to the proposition just proved, either a largest term γ or all its terms αv would be exceeded by a specific number β from (II). In the first case the number γ + 1 belongs to the second number-class (II), while in the second case the number β would belong to the second number-class (II) and could not appear in the sequence (αv), otherwise, by the already assumed properties of the sets (II) and (αv), there would be a contradiction. Therefore the second number-class (II) has a different cardinal-number than the first number-class (I).

For the two cardinal-numbers of the number-classes (I) and (II) it is really the case that the second is the cardinal-number that follows the first infinite cardinal-number, that is, there are no other cardinal-numbers between these two cardinal-numbers, and this fact emerges with absolute certainty from a proposition which I will shortly state and prove.

However, if we first take a look backwards at the means that have led both to an expansion of the concept of integers and to a new collection of different but well-defined sets, we observe that there were three prominent, but quite distinct logical principles. There are both the two principles of generation defined above and then also the additional principle of limitation or restriction, which consists of the requirement that the creation of a new integer using one of the other two principles is permitted only if the set of all preceding numbers has the cardinal-number of an already existing, defined number-class. In this way, by observing these three principles, one can arrive at ever new sets of numbers with the greatest logical certainty, and with them all the different, successively ascending cardinal-numbers that occur in the physical and conceptual realms, and the new numbers obtained also have the same concrete determinateness and objective reality as the earlier ones; I therefore really do not know what should hold us back from this activity of forming new numbers, once it appears that the introduction of a new one of these innumerable number-classes has become desirable or even indispensable for the advancement of science.