Ordinal-numbers

It will now be shown how we progress to the definition of the new numbers, and how the natural segmentation into what I call number-classes results from the absolutely infinite sequence of actual integers. To this discussion I will only add the main propositions about the second number-class and its relation to the first.

The sequence (I) of positive actual integers 1, 2, 3, …, v, … has its origin in the repeated positioning and combination of elemental units that are regarded as all identical; the number v is the expression both for a definite finite number of such successive positions, as well as for the union of the positioned units into a whole. The formation of finite integers is based on the principle of adding a unit to an existing number that has already been formed;

I call this principle, which, as we shall soon see, also plays an essential role in the generation of the higher integers, the first principle of generation. The cardinal-number of the numbers v of the number-class (I) which is formed in this way is infinite; there is no largest one of these numbers v.

As contradictory as it would be to speak of a largest number in the number-class (I), on the other hand it is not abhorrent to think of a new number, which we want to define as ω,[Footnote25] and which is to be an expression for the fact that the natural succession of the complete number-class (I) is given according to established rules (similarly, v is an expression for the fact that a certain finite number of units are combined into a whole).

It is even permissible to think of the newly created number ω as the limit towards which the numbers v approach, provided nothing else is understood by that, so that ω is to be the first integer that follows all numbers v, i.e: is to be called larger than each of the numbers v. By allowing further positions of the unit to follow the positioning of the number ω, further numbers are obtained with the aid of the first principle of generation:

ω + 1, ω + 2, …, ω + v, …

Since again one does not arrive at a largest number here, one conceives of a new one, which one calls 2ω and which should be the first one that follows all previous numbers v and ω + v. If one applies the first principle of generation repeatedly to the number 2ω, one arrives at the continuation:

2ω + 1, 2ω + 2, …, 2ω + v, …

of the previous numbers.

This logical operation, which has given us the two numbers ω and 2ω, is obviously different from the first principle of generation; I shall call this the second principle of generation of integers and I define it more precisely thus: If there is any definite succession of defined integers of which there is no largest integer, a new number is created on the basis of this second principle of generation, which is thought of as the limit of those numbers, i.e: defined as the next number larger than all of them.

Through the combined application of both principles of generation one thus successively obtains the following continuations of our numbers obtained so far:

3ω + 1, 3ω + 2, …, 3ω + v, …

…, …, …, …

μω + 1, μω + 2, …, μω + v, …

…, …, …, …

However, this does not lead to an endpoint either, because none of the numbers μw + v is a largest number.

The second generation principle therefore prompts us to introduce the next number that follows all numbers μw + v, which can be called ω2, and which is followed by the definite succession numbers:

λω + μω + v

and following the two principles of generation it is then apparent that we arrive at numbers of the following form:

v0ωμ + v0ωμ-1 + … + vμ-1ω + vμ

but then the second principle of generation drives us to define a new number, which is to be the next largest of these numbers and which is referred to by:

ωω

As one can see, there is no end to the formation of new numbers. If both principles of generation are followed, one always obtains new numbers and sequences of numbers that have a completely determined succession.

One is thereby initially given the impression that this way of forming new whole definitive-infinite numbers would cause us to lose ourselves in limitlessness, and that we would be unable to give this endless process a certain temporary termination, in order to thereby impose a similar restriction point, as it actually exists in a certain sense in relation to the first number-class (I); only the first principle of generation was used there, making it impossible to step out of that sequence (I).

The second principle of generation is required to not only lead beyond the previous numerical domain, but also proves to be a means by which, in conjunction with the first principle of generation, we are enabled to break through every barrier in the formation of the concept of integers.

But if we now note that all the numbers obtained so far, and those immediately following them, satisfy a certain property, then this property provides us with a new third principle in addition to these two, if we impose it as a condition on the formation of all new numbers.[Footnote26] I call this the principle of limitation or restriction which, as I will show, has the effect that the second number-class (II), defined by the inclusion of this third principle, not only obtains a greater cardinal-number than that of the first number-class (I), but precisely the next higher cardinal-number, the second cardinal-number.

One is at once convinced that the property mentioned, which is satisfied by each of the infinite numbers α previously defined - that the set of the numbers preceding this number in the sequence of numbers has the cardinal-number of the first number-class (I). Take for example, the number ωω; the preceding ones are included in the formula:

v0ωμ + v0ωμ-1 + … + vμ-1ω + vμ

where μ, v0, v1, …, vμ are all finite positive integers, including zero but excluding the case where v0 = v1 = … = vμ = 0.

As is well known, this set can be brought into the form of a simple infinite sequence and thus has the cardinal-number of the first number-class (I).

For every sequence of sets, where each set is of the first infinite cardinal-number, then provided that sequence itself is of the first infinite cardinal-number, it always results in a set which has the cardinal-number of the first number-class (I), and it is clear that if we continue our sequence of numbers we only ever obtain numbers which actually satisfy that condition.

We therefore define the second number-class (II) as the set of all numbers α that can be formed with the aid of the two principles of generation and which progress in a definitive succession:

ω, ω + 1, …, v0ω + v0ωμ-1 + … + vμ-1ω + vμ, …, ωω, …, α, …

which are subject to the condition that all of the numbers preceding α, from 1 on, form a set with the cardinal-number of the first number-class (I).